M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
Applications of Calculus to the Physical World
1)!
2U84-9i
Farmer Brown packs her unsalted butter in rectangular cardboard boxes with square ends and
overlapping lids which exactly cover the open top, the square ends, and the front face. Each
box uses 0·27 square metres of cardboard. What is the maximum volume of a box?¤
« 0·0054 m3 »
2)!
2U84-10i
A particle P moves along a straight line so that at time t ( 0) its displacement from a fixed
point O on that line is given by x(t) = 3t2(4 + t3)1.
a.
Find the velocity of the particle at time t.
b.
Find the times when the particle is momentarily at rest.
c.
Show that P is in (exactly) the same position at both the times t1 = 1, t 2  2  2 2 .
Give a brief description of the way the displacement changes over the time period
from t1 to t2.
d.
Describe the motion of the particle as t increases without bound.
e.
Find the maximum displacement from O.¤
3 t (8  t 3)
« a)
b) t = 0, 2 c) When t = 1 the particle is at x = 3 . It then moves to x = 1 when
3 2
5
(4  t )
t = 2 before returning to x = 3 when t = 2  2 2 . d) As t approaches , the particle
5
approaches the origin - its velocity decreasing as it does so. e) 1 unit in the positive
direction »
3)!
2U85-8ii
An open rectangular box has four sides and a base, but no lid, as in the figure.
y cm
x cm
a.
b.
2x cm
The dimensions of the base of the box are x cm, 2x cm, and the height is y cm.
Write down formulae for the area A cm2 of the outer surface of the box, and the
volume V cm3 contained by the box.
Given that A = 150, eliminate y to obtain a formula V(x) for the volume as a
function of x. Hence
.
show that x  5 3 .
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
304
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
.
4)!
find the value of x for which V is maximum, and verify that the maximum
500
value of V is
.¤
3
« a) A = 2x2 + 6xy, V = 2x2y b) ) Proof ) x = 5, Proof »
2U85-9i
t minutes after a jet engine starts operation, the rate of fuel burn, R kg per minute, is given by
10
the relation R  10 
.
1  2t
a.
Draw a sketch of R as a function of t.
b.
What is the rate of burn, R, after 7 minutes?
c.
What value does R approach as t becomes very large?
d.
Calculate the total amount of fuel burned in the first 7 minutes.¤
R
20
10
t
« a)
5)!
6)!
b) 10
2
kg/min c) 10 d) 83·5 kg (to 1 d.p.) »
3
2U85-9ii
The acceleration a metres per second per second of a moving object is given at time t seconds
(t  0) by a = 4cos t. At time t = 0, the object is at the point x = 0, and travelling with
velocity V = 2 metres per second.
a.
Find the velocity V and the displacement x as a function of t, for t  0.
b.
Find, for t in the range 0  t  4, the values of t for which the object is stationary.
19
c.
Show that, for t in the range 0  t  4, the largest value of x is 2(2  3 ) 
.¤
3
« a) V = 4 sin t + 2, x = 4 cos t + 2t + 4 b) t = 7 , 11 , 19 , 23 seconds c) Proof »
6 6 6 6
2U85-10ii
A population N(t) varies with time according to the law N(t) = Cekt, where C, k are positive
constants and t  0.
a.
Show that, if a, b are two positive numbers such that a + b = 1, then
N (at + bu ) = N (t )a N (u )b for any t  0, u  0.
b.
Hence, or otherwise, find N(13), given that N(3) = 10 and N(18) = 100.¤
5
3
« a) Proof b) 10 »
7)!
8)!
2U86-5i
The displacement x metres from the origin at time t seconds, of a particle travelling in a
straight line is given by the formula x = t3  21t2.
a.
Find the acceleration of the particle at time t seconds.
b.
Find the times at which the particle is stationary.¤
« a) (6t  42) ms2 b) t = 0, 14 seconds »
2U86-9ii
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
305
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
B
r metres
A

C
9)!
10)!
In the figure, AB and AC are radii of length r metres, of a circle with centre A. The arc BC of
the circle subtends an angle  radians at A.
a.
Write down formulae for
.
the length of the arc BC,
.
the area of the sector ABC.
b.
The perimeter of the above figure ABC (i.e. the length AB + arc BC + CA) is
12 metres. Show that the area Y square metres of the sector ABC is given by
72θ
y=
. Hence show that the maximum area of the sector is 9 square metres.¤
(θ + 2 )2
1
« a) ) L = r ) A = r2 b) Proof »
2
2U86-10i
The rate of increase of a population P(t) of persons in a certain country is governed by the
dP
 kP where k is a constant, and t is the time in years. The population of the
equation
dt
country doubles every twenty years.
a.
Find k.
b.
In which year will the country reach a population three times that it had at the
beginning of 1960?
c.
Given that at the beginning of 1960 the population was 15·1 million, what will be
the population at the beginning of the year 2010?¤
1
« a)
loge2 b) 1991 c) 8·54  107 (to 3 sig. figs) »
20
2U87-8ii
A particle moves in a straight line. At time t seconds its displacement x metres from a fixed
point O on the line is given by x = 3  cos 2t, 0  t  2.
a.
Sketch the graph of x as a function of t.
b.
Write down the times when the particle is at rest and the position of the particle at
each of these times.
c.
Find the time when the particle first reaches its maximum speed.¤
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
306
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
x
4
2
« a)
11)!
12)!
π
2

3π
2
2
t
b) The particle is at rest at t = 0,  , 3 and 2 seconds. Its
2 2
displacement at these times is x = 2, 4, 2, 4, and 2 m, respectively. c)  seconds »
4
2U87-9ii
Tom took a bottle which had 200 millilitres of water in it. He poured more water into it for
twenty seconds until it was full. During this time the volume flow rate R of water, in
millilitres per second, into the bottle was given by R = 4(20  t).
a.
Find a formula for the volume V of water in the bottle after t seconds where t  20.
b.
How many millilitres of water were in the bottle when it was full?
c.
How long did it take before the bottle was half full?¤
« a) V = 200 + 80t  2t2 b) 1000 mL c) 4·2 seconds »
2U87-10ii
A children's picture book is being designed so that each page contains 320 square centimetres
of print and pictures surrounded completely by a white border as illustrated in the figure
below.
x cm
3 cm
y cm
WHITE
PRINT
BORDER
2 cm and 2 cm
PICTURES
Not to Scale
2 cm
13)!
Each page is to have a border of width 2 centimetres at the bottom and on each side, as well
as a border of width 3 centimetres at the top. Let the width of a page be x centimetres and its
length be y centimetres.
a.
Show that the area A square centimetres of one such page is given by
320
A  x (5 
).
(x  4)
b.
Show that the page which fulfills all the printing requirements and which has the
smallest area is 20 centimetres wide and 25 centimetres long.¤
« Proof »
2U88-4b
The displacement x metres from the origin at time t seconds of a particle travelling in a
straight line is given by x = t3  9t, where t  0.
i.
Find the velocity at time t seconds.
ii.
Calculate the velocity when t = 2.
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
307
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
iii.
14)!
Find the time when the particle is stationary.¤
« i) 3t2  9 m/s ii) 3 m/s iii)
3 seconds »
2U88-9a
y
6y
x Pig PADDOCK
Pen
FOR
Pig
x
Pen CALVES
BRICK WALL
Farmer Brown wishes to construct three rectangular enclosures, as shown above, in which to
put pigs and calves. The paddock for the calves is to be six times as long and twice as wide
as a pig pen. One pig pen and the calves' paddock have an existing brick wall as a boundary
fence as shown. All other fences are to be constructed from 56 metres of wire mesh.
i.
Let x metres be the width of a pig pen and y metres be its length. Show that
3
y 7  x .
4
ii.
Hence show that the total area A square metres contained in the three enclosures is
3
given by A  14 x(7  x) .
4
iii.
Show that A is a maximum when half the wire fencing has been placed parallel to
the brick wall.¤
« Proof »
15)! 2U88-10a
The councils in two towns A and B have found that the populations in the towns are given by:
PA = 2000e0.02t for town A, PB = 1000e0.03t for town B, where t is the number of years which
have elapsed since January 1st, 1980.
i.
Write down the annual growth rate for B.
ii.
Calculate the instantaneous rate at which the population of B will be increasing at
the start of 1995.
iii.
During which year will the population of B become larger than the population
of A?¤
« i) 3% ii) Approx. 47 people in 1995 iii) 1993 »
16)! 2U89-5c
A particle moves in a straight line so that its velocity v in metres per second at time t is given
by v = 4  2t. At time t = 0, the particle is at x = 1.
i.
Find the displacement x of the particle as a function of t.
ii.
When is the particle at rest and what is its acceleration at that time?
iii.
Find the distance the particle travels in the first 4 seconds.¤
« i) x = 4t  t2 + 1 ii) The particle is at rest when t = 2 secs and has an acceleration of 2 m/sec2.
iii) 8 m »
17)! 2U89-7b
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
308
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
The rate of emission E, in tonnes per year, of chloro-fluorocarbons (CFCs) in Australia from
2
40
) , where t is the time in years.
31 October 1989 will be given by E  100  (
1t
i.
What is the rate of emission E on 31 October 1989?
ii.
What is the rate of emission E on 31 October 1998?
iii.
What value does E approach as the years pass by?
iv.
Draw a sketch of E as a function of t.
v.
Calculate the total amount of CFCs emitted in Australia during the years 1989 to
1998.¤
« i) 1700 tonnes per year ii) 116 tonnes per year iii) 100 tonnes per year
E
1700
100
t
iv)
18)!
v) 2340 tonnes »
2U89-10b
A cam is formed with cross-section as shown in the figure.
B
r
A

O
r
2
r
2
C
X
The cross-section consists of a semi-circle AXC centre O and radius
19)!
r
and a sector ABC of
2
radius r, centre A and angle .
i.
What is the perimeter ABCX of the cam in terms of r and ?
ii.
If the area of the cross-section of the cam is 1 square unit, show that the perimeter P
2

is given by P   r (1  ) .
r
4
8
iii.
Show that the least perimeter occurs when r 2 =
and calculate the value of 
 4
to the nearest degree.¤
1
« i) (r + r + r) units ii) Proof iii) 57 »
2
2U90-6b
A particle P is at the origin at time t = 0 and moves so that its velocity for t  0 is given by
1
v
.
t 1
i.
What is the acceleration of P when t = 1?
ii.
What is the displacement x of P from the origin when t = 1?¤
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
309
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
« i)  1 ii) ln 2 »
4
20)!
21)!
22)!
2U90-7c
A quantity Q of radium at time t in years is given by Q = Q0ekt where k is a constant and Q0
is the initial amount of radium at time t = 0.
i.
Given that Q = 1 Q0, when t = 1690 years, calculate k.
2
ii.
After how many years does only 20% of the initial amount of radium remain?¤
« i) 4·1014  104 ii) 3924 years »
2U90-8c
It is assumed that the number N(t) of termites in a certain mound at time t  0 is given by
A
, where A is a constant, and t is measured in months.
N (t ) 
2  e t
i.
At time t = 0, N(t) is estimated at 3  105 termites. What is the value of A?
ii.
What is the value of N(t) after one month?
iii.
How many termites would you expect to find in the mound when t is very large?
iv.
Find an expression for the rate at which the number of termites increases at any
time t.¤
t
« i) 9  105 ii) 380 087 iii) Approximately 450 000 iv) Ae t 2 »
(2  e )
2U90-9b
20 cm
h
r
6 cm
23)!
A cylinder of radius r cm and height h cm is inscribed in a cone with base radius 6 cm and
height 20 cm as in the diagram.
2
10 r (6  r )
i.
Show that the volume V of the cylinder is given by V =
.
3
ii.
Hence find the values of r and h for the cylinder which has maximum volume.
iii.
What is the maximum volume?¤
320
« i) Proof ii) r = 4, h = 20 iii)
cm3 »
3
3
2U91-7b
The amount A grams of a given carbon isotope in a dead tree trunk is given by A = A0ekt
where A0 and k are positive constants, and where the time t is measured in years from the
death of the tree.
dA
  kA .
i.
Show that A satisfies the equation
dt
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
310
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
ii.
iii.
Find the value of k if the amount of isotope is halved every 5500 years.
For a particular dead tree trunk the amount of isotope is only 15% of the original
amount in the living tree. How long ago did the tree die? Give your answer to the
nearest 1000 years.¤
« i) Proof ii) 1·26  104 iii) 15 000 years »
24)!
2U91-8b
A particle moves along a straight line so that its distance x, in metres, from a fixed point O is
given by x = 1  2 sin 2t where the time t is measured in seconds from t = 0.
i.
Where is the particle initially?
ii.
When, and where, does the particle first come to rest?
iii.
Where does the particle next come to rest?

iv.
What is the acceleration of the particle when t 
?¤
12
3

« i) 1 m to the right of O ii) t = , 1 m to the left of O iii) 3 m to the right of O (at t =
)
4
4
iv) 4 ms2 »
25)! 2U91-10b
2 km
x km
A
P
B
1 km
O
The diagram shows a straight section of a river, one kilometre wide. Adrienna is at a point O
on one bank and she wishes to reach a point B on the opposite bank. The point A is directly
opposite O and the distance from A to B is two kilometres. Adrienna can row at 6 km/h and
jog at 10 km/h. She intends to row in a straight line to a point P on the opposite bank and
then jog directly from P to B. Let the distance AP be x kilometres.
i.
Show that the time T, in hours, that Adrienna takes to reach B is given by
ii.
26)!
27)!
2
1 2  x
.
T x

6
10
Show that if Adrienna wishes to minimize the time taken to complete the journey
then she should row to a point P, 3 kilometre from A.¤
4
« Proof »
2U92-4c
Ten kilograms of sugar is placed in a container of water and begins to dissolve. After t hours
the amount A kg of undissolved sugar is given by A = 10ekt.
i.
Calculate k, given that A = 3·2 when t = 4.
ii.
After how many hours does 1 kg of sugar remain undissolved?¤
« i) k  0·28 (to 2 d.p.) ii) 8 hrs (to nearest hr) »
2U92-6c
The speed of a train was recorded at intervals of one minute. The times, in minutes, and the
corresponding speeds v, in kilometres per hour, are listed in the table below.
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
311
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
time (min)
v (km/h)
i.
0
0
1
25
2
34
3
30
4
40
Explain why the distance x, in km, travelled by the train in these four minutes is
1
15
given by x   v dt .
0
ii.
Estimate x by using Simpson's Rule with five function values.¤
b
« i) By definition x   v dt . Since the units of velocity are km/h, the limits of integration are
a
t = 0 hrs and t = 1 hrs (4 mins). ii) 1·8 km (to 1 d.p.) »
15
28)!
2U92-7c
x
t
t4
t1
t2
t3
0
A particle moves in a straight line and the above graph shows the distance x of the particle
from a fixed point at time t.
i.
Is the particle moving faster at time t1 or at time t2? Why?
ii.
What is the velocity at time t = 0? Why?
iii.
Sketch the graph of the velocity v as a function of t.¤
« i) The particle is moving faster at t2, since the gradient of the curve is greater at t2 than at t1.
ii) The velocity is zero since the curve is horizontal (i.e. the gradient is zero at t = 0).
v
t1
t2
t3
t4
iii)
29)!
2U92-8a
Twenty-five kangaroos were released on an island. The population P of kangaroos on the
island t years later is given by P = t3 + 6t2 + 25, for 0  t  6.
i.
After how many years was the population a maximum?
ii.
What was the maximum population?
iii.
Sketch the curve P = t3 + 6t2 + 25, for 0  t  6.
iv.
When was the population increasing most rapidly?¤
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
312
t
»
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
P
(0, 25)
(4, 57)
(6, 25)
t
« i) 4 years ii) 57 kangaroos iii)
iv) At t = 2 years »
30)! 2U92-10b
A truck is to travel 1000 kilometres at a constant speed of v km/h. When travelling at v km/h,
2
the truck consumes fuel at the rate of (6  v ) litres per hour. The truck company pays
50
50 cents/litre for fuel and pays each of the two drivers $20 per hour whilst the truck is
travelling.
i.
Let the total cost of fuel and the drivers' wages for the trip be C dollars. Show that
1
C  10 v  43 000 .
v
ii.
The truck must take no longer than 12 hours to complete the trip, and speed limits
require that v  100. At what speed v should the truck travel to minimize the
cost C?¤
1
« i) Proof ii) 83 km/h »
3
31)! 2U93-3b
'Although the amount of pollution is increasing, the Government's policies to reduce
pollution seem to be taking effect.'
dP
Given that P is the amount of pollution, what does the above statement imply about
and
dt
2
d P
?¤
2
dt
dP
d2 P
« Although pollution is increasing (i.e.
> 0), the rate of increase is slowing (i.e.
< 0). »
dt
dt 2
32)! 2U93-4b
Under certain climatic conditions the number N of blue-green algae satisfies the equation
N (t )  Ae0.15t where t is measured in days and A is a constant.
i.
Show that the number of algae increases at a rate proportional to the number present.
ii.
When t = 3 the number of algae was estimated to be 1·7  108. Evaluate A.
iii.
The number of algae doubles every x days. Find x.¤
« i) Proof ii) 1·083 968  108 iii) 4·621 (to 3 d.p.) »
33)! 2U93-9a
'The car moved away from where it had stopped, its speed increasing at a constant rate, and
after exactly 10 seconds it was travelling at 25 m/s. It continued at a constant speed for a
further 20 seconds. Then the brakes were applied causing it to slow down at a constant rate,
so that 5 seconds later it was travelling at 5 m/s.
i.
Let the car's speed be v m/s. Graph v as a function of time t, measured in seconds.
ii.
Let the distance travelled by the car be s metres from where it had stopped. On a
separate diagram, graph s as a function of time t.¤
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
313
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
v
25
s
685·5
625
5
125
10
« i)
34)!
10
30 35 t ii)
30 35 t
»
2U93-9b
y
1·73 m
1·35 m
t
35)!
0:00 hours
12·42 hours
(midnight
1) Jan. 1990
The diagram shows the tidal effect due to the Moon at Port Hedland on 1 January 1900. The
water level can be approximated by a sine curve of the form y = A sin(at + b) where y is the
water level in metres measured as on the diagram and t is the time in hours after 0:00 hours.
i.
Find the amplitude A.
ii.
Estimate b by letting t = 0.
iii.
Estimate a.¤
« i) 1·73 m ii) b = 0·8952 (to 4 d.p.) iii) a = 0·5059 (to 4 d.p.) »
2U93-10b
y
2
y = x + 2 B(2, 4)
y=x
A(–1, 1)
36)!
P(t, t2)
x
O
In the diagram A(1, 1) and B(2, 4) are the points of intersection of the parabola y = x2 with
the line y = x + 2. The point P(t, t2) is a variable point on the parabola below the line.
i.
Find the area of the parabolic segment APB, i.e. the area below the line and above
the parabola.
ii.
Show that the maximum area of triangle APB is three-quarters of the area of the
parabolic segment APB.¤
1
« i) 4 units2 ii) Proof »
2
2U94-5c
A particle moves along a straight line so that its distance x metres from a fixed point O is
given by x = 6  2t + 8 ln(t + 3), where the time t is measured in seconds.
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
314
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
i.
ii.
iii.
37)!
What is the position of the particle when t = 0?
Find expressions for the velocity and acceleration of the particle at time t.
Find the time t when the velocity of the particle is zero.¤
8
8
« i) 14·8 m (to 1 d.p.) ii) v =  2 
, a= 
iii) t = 1 second »
t 3
(t  3 )2
2U94-7a
r
Baked
Beans
38)!
39)!
h
A can of baked beans is in the shape of a closed cylinder with height h cm and radius r cm, as
shown in the diagram.
i.
The volume of the can is 500 cm3. Find an expression for h in terms of r.
1000
ii.
Show that the surface area, S cm2, of the can is given by S  2 r 2 
.
r
iii.
If the area of metal used to make the can is to be minimized, find the radius of the
can.¤
500
250
3
« i) h =
ii)
Proof
iii)
 4·3 cm (to 1 d.p.) »

 r2
2U94-7c
A Geiger counter is taken into a region after a nuclear accident and gives a reading of 10 000.
One year later, the same Geiger counter gives a reading of 9000. It is known that the reading
T is given by the formula T = T0ekt, where T0 and k are constants and t is the time measured
in years.
i.
Evaluate the constants T0 and k.
ii.
It is known that the region will become safe when the reading reaches 40. After how
many years will the region become safe?¤
« i) T0 = 10 000, k = ln 9 ii) Approximately 52·4 years »
10
2U94-8a
The number of unemployed people u at time t was studied over a period of time. At the start
of this period, the number of unemployed was 800 000.
du
< 0 . What does this say about the number of
i.
Throughout the period,
dt
unemployed during the period?
2
u
ii.
It is also observed that, throughout the period, d 2 > 0. Sketch the graph of u
dt
against t.¤
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
315
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
u
800 000
40)!
41)!
t
« i) The number of unemployed was decreasing ii)
»
2U95-6c
Coal is extracted from a mine at a rate that is proportional to the amount of coal remaining in
the mine. Hence the amount R remaining after t years is given by R = R0ekt, where k is a
constant and R0 is the initial amount of coal. After 20 years, 50% of the initial amount of coal
remains.
i.
Find the value of k.
ii.
How many more years will elapse before only 30% of the original amount
remains?¤
« i) 0·03466 (to 4 s.f.) ii) Approx. 14·7 years »
2U95-7c
Level of
pollutant
1945
42)!
Time
1995
The graph shows the levels of a pollutant in the atmosphere over the past 50 years.
Describe briefly how the level of this pollutant has changed over this period of time. Include
mention of the rate of change.¤
« The level of pollutant is increasing, but the rate of increase is slowing. »
2U95-9a
w
15
d
A rectangular beam of width w cm and depth d cm is cut from a cylindrical pine log as
shown. The diameter of the cross-section of the log (and hence the diagonal of the crosssection of the beam) is 15 cm. The strength S of the beam is proportional to the product of its
width and the square of its depth, so that S = kd2w.
i.
Show that S = k(225w  w3).
ii.
What numerical dimensions will give a beam of maximum strength?
Justify your answer.
iii.
A square beam with diagonal 15 cm could have been cut from the log.
Show that the rectangular beam of maximum strength is more than 8% stronger than
this square beam.¤
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
316
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
« i) Proof ii) 5 3 cm  5 6 cm iii) Proof »
43)!
2U95-9b
v
(velocity)
10
0
t (time)
7
A particle is observed as it moves in a straight line in the period between t = 0 and t = 10. Its
velocity v at time t is shown on the graph above. Copy or trace this graph into your Writing
Booklet.
i.
On the time axis, mark and clearly label with the letter Z the times when the
acceleration of the particle is zero.
ii.
On the time axis, mark and clearly label with the letter G the time when the
acceleration is greatest.
iii.
There are three occasions when the particle is at rest, i.e. t = 0, t = 7, and t = 10.
The particle is furthest from its initial position on one of these occasions. Indicate
which occasion, giving reasons for your answer.¤
v
Turning point
Inflexion
Z
G Z
« i) ii)
44)!
t
iii) t = 7 »
2U95-10a
i.
Draw the graphs of y = 4 cos x and y = 2  x on the same set of axes for
2  x  2.
ii.
Explain why all the solutions of the equation 4 cos x = 2  x must lie between
x = 2 and x = 6.¤
y
y = 4 cos x
4
2
2
–
 x2 2
x3
x1
x
4
« i)
45)!
7
10
Turning
point
y=2x
ii) Proof »
2U95-10b
Two particles A and B start moving on the x-axis at time t = 0. The position of particle A at
1
time t is given by x = 6 + 2t  t2 and the position of particle B at time t is given by
2
x = 4 sin t.
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
317
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
i.
ii.
iii.
iv.
46)!
Find expressions for the velocities of the two particles.
Use part (a) to show that there are exactly two occasions, t1 and t2, when these
particles have the same velocity.
Show that the distance travelled by particle A between these two occasions is
1
4  2(t1 + t2) + (t12 + t22).
2
Show that the two particles never meet.¤
« i) VA = 2  t, VB = 4 cos t ii) iii) iv) Proof »
2U96-6a
Lee takes some medicine. The amount, M, of medicine present in Lee’s bloodstream t hours
later is given by M = 4t2  t3, for 0  t  3.
i.
Sketch the curve M = 4t2  t3, for 0  t  3 showing any stationary points.
ii.
At what time is the amount of medicine in Lee’s bloodstream a maximum?
iii.
When is the amount present increasing most rapidly?¤
M ( 2 2 , 9 13 )
3
27
t ii) 2 hours 40 mins iii) After 1 hour 20 mins. »
« i)
47)!
2U96-8b
T
x cm
P
6 cm
Q
4 cm
R y cm U
S
48)!
PQRS is a rectangle with PQ = 6 cm and QR = 4 cm. T and U lie on the lines SP and SR
respectively, so that T, Q and U are collinear, as shown in the diagram. Let PT = x cm and
RU = y cm.
i.
Show that triangles TPQ and QRU are similar.
ii.
Show that xy = 24.
48
iii.
Show that the area, A, of triangle TSU is given by A = 24 + 3x +
.
x
iv.
Find the height and base of the triangle TSU with minimum area. Justify your
answer.¤
« i) ii) iii) Proof iv) height = 8 cm, base = 12 cm »
2U96-9
Two particles P and Q start moving along the x axis at time t = 0 and never meet. Particle P
is initially at x = 4 and its velocity v at time t is given by v = 2t + 4.
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
318
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
The position of particle Q is given by x = 1 + 3 loge(t + 1). The diagram shows the graph of
x = 1 + 3 loge(t + 1).
x
x = 1 + 3 loge(t + 1)
1
1 2 t
Find an expression for the position of P at time t.
Copy the diagram into your Writing Booklet and, on the same set of axes, draw the
graph of the function found in part (a).
P and Q are joined by an elastic string and M is the midpoint of the string. Show
1
that the position of M at time t is given by x = [t2 + 4t + 3 loge(t + 1) + 5].
2
Find the time at which the acceleration of M is zero.
Find the minimum distance between P and Q.¤
x
x = t2 + 4t +4
0
a.
b.
c.
d.
e.
4
« a) x = t2 + 4t + 4 b)
49)!
x = 1 + 3 loge(t + 1)
1
0
1 2
c) Proof d) 1 +
t
6
e) 3 units »
2
2U97-6a
P
Q
50)!
R
x metres
A wire of length 5 metres is to be bent to form the hypotenuse and base of a right-angled
triangle PQR, as shown in the diagram. Let the length of the base QR be x metres.
i.
What is the length of the hypotenuse PQ in terms of x?
1
ii.
Show that the area of the triangle PQR is x 25  10x square metres.
2
iii.
What is the maximum possible area of the triangle?¤
« i) (5  x)m ii) Proof iii) 2·4 m2 (to 1 dp) »
2U97-7c
A ball is dropped into a long vertical tube filled with honey. The rate at which the ball
dv
  kv , where v is the velocity in
decelerates is proportional to its velocity. Thus
dt
centimetres per second, t is the time in seconds, and k is a constant. When the ball first enters
the honey, at t = 0, v = 100. When t = 0·25, v = 85.
dv
  kv .
i.
Show that v = Cekt satisfies the equation
dt
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
319
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
ii.
iii.
iv.
51)!
52)!
Find the value of the constant C.
Find the value of the constant k.
Find the velocity when t = 2.¤
« i) Proof ii) 100 iii) 0·65 (to 2 dp) iv) 27·25 cm/s (to 2 dp) »
2U98-5b
The population P of a city is growing at a rate that is proportional to the current population.
The population at time t years is given by P = Aekt, where A and k are constants. The
population at time t = 0 was 1 000 000 and at time t = 2 was 1 072 500.
i.
Find the value of A.
ii.
Find the value of k.
iii.
At what time will the population reach 2 000 000?¤
1
« i) 1 000 000 ii) ln ( 1  0725 ) iii) 19·8 years »
2
2U98-6a
A particle P moves along a straight line for 8 seconds, starting at the fixed point S at time
t = 0. At time t seconds, P is x(t) metres to the right of S. The graph of x(t) is shown in the
diagram.
x
0
i.
ii.
iii.
iv.
53)!
54)!
3
6
t
At approximately what times is the velocity of the particle equal to 0?
At approximately what time is the acceleration of the particle equal to 0?
At approximately what time is the distance from S greatest?
At approximately what time is the particle moving with the greatest velocity?¤
« i) t = 1 and t = 5·4 secs ii) t = 3 secs iii) t = 5 secs iv) t = 8 secs »
2U98-8a
Sand is tipped from a truck onto a pile. The rate, R kg/s, at which the sand is flowing is given
by the expression R = 100t – t3, for 0  t  T, where t is the time in seconds after the sand
begins to flow.
i.
Find the rate of flow at time t = 8.
ii.
What is the largest value of T for which the expression for R is physically
reasonable?
iii.
Find the maximum rate of flow of sand.
iv.
When the sand starts to flow, the pile already contains 300 kg of sand. Find an
expression for the amount of sand in the pile at time t.
v.
Calculate the total weight of sand that was tipped from the truck in the first
8 seconds.¤
2000 3
1
kg/s iv) 50t2 – t 4  300 v) 2176 kg »
« i) 288 kg/s ii) T = 10 iii)
4
9
2U99-6a
The mass M kg of a radioactive substance present after t years is given by M = 10e–kt, where
k is a positive constant. After 100 years the mass has reduced to 5 kg.
i.
What was the initial mass?
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
320
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
ii.
iii.
iv.
Find the value of k.
What amount of the radioactive substance would remain after a period of
1000 years?
How long would it take for the initial mass to reduce to 8 kg?¤
« i) 10 kg ii) 0·00693 (3 sig. figs) iii) 9·766 g iv) 32·2 years »
55)!
2U99-7b
A particle P is moving along the x axis. Its position at time t seconds is given by:
x = 2 sin t – t, t  0.
i.
Find an expression for the velocity of the particle.
ii.
In what direction is the particle moving at t = 0?
iii.
Determine when the particle first comes to rest.
iv.
When is the acceleration negative for 0  t  2?
v.
Calculate the total distance travelled by the particle in the first  seconds.¤


« i) v = 2 cos t – 1 ii) The positive direction iii) t = seconds iv) 0 < t <  v) 2 3 
»
3
3
56)! 2U99-9b
S
V
R
3m
T
Not to scale
ym
U
Q
4m
In the diagram, PQ and SR are parallel railings which are 3 metres apart. The points P and Q
are fixed 4 metres apart on the lower railing. Two crossbars PR and QS intersect at T as
shown in the diagram. The line through T perpendicular to PQ intersects PQ at U and SR at
V. The length of UT is y metres.
SR VT
i.
By using similar triangles, or otherwise, show that
.

PQ UT
12
ii.
Show that SR 
4.
y
18
iii.
Hence show that the total area A of  PTQ and  RTS is A  4 y  12 
.
y
iv.
Find the value of y that minimises A. Justify your answer.¤
3 2
« i) ii) iii) Proof iv)
»
2
2U00-5c
The population of a certain insect is growing exponentially according to N = 200ekt, where t
is the time in weeks after the insects are first counted. At the end of three weeks the insect
population has doubled.
i.
Calculate the value of the constant k.
ii.
How many insects will there be after 12 weeks?
iii.
At what rate is the population increasing after three weeks?¤
« i) 0·2310 (to 4 d.p) ii) 3200 (to 2 sig. figs) iii) 92·4 insects per week (to 2 sig. figs) »
P
57)!
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
321
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
58)!
2U00-6b
The number N of the students logged onto a website at any time over a five-hour period is
approximated by the formula N = 175 + 18t2 – t4, 0  t  5.
i.
What was the initial number of students logged onto the website?
ii.
How many students were logged onto the website at the end of the five hours?
iii.
What was the maximum number of students logged onto the website?
iv.
When were the students logging onto the website most rapidly?
v.
Sketch the curve N = 175 + 18t2 – t4 for 0  t  5.¤
(3, 256)
N
175
59)!
60)!
( 3 , 220)
0 1
t
2 3 4
5
« i) 175 ii) 0 iii) 256 iv) t  3 hours v)
»
2U00-8a
A particle is moving in a straight line, starting from the origin. At time t seconds the particle
has a displacement of x metres from the origin and a velocity v ms–1. The displacement is
given by
x = 2t – 3 loge(t + 1)
i.
Find an expression for v.
ii.
Find the initial velocity.
iii.
Find when the particle comes to rest.
iv.
Find the distance travelled by the particle in the first three seconds.¤
1
3
« i) v  2 
ii) –1 m/s iii) t  sec iv) 1·84 m (to 3 sig. figs) »
2
t 1
2U00-8b
An enclosure is to be built adjoining a barn, as in the diagram. The walls of the barn meet at
135, and 117 metres of fencing is available for the enclosure, so that x + y = 117 where
x and y are as shown in the diagram.
x
Enclosure Barn
y
135
Barn
i.
ii.
Show that the shaded area of the enclosure in square metres is given by
3
A  117 x  x 2 .
2
Show that the largest area of the enclosure occurs when y = 2x.¤
« Proof »
61)!
2U00-10b
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
322
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
62)!
63)!
64)!
65)!
The first snow of the season begins to fall during the night. The depth of the snow, h,
increases at a constant rate through the night and the following day. At 6 am a snow plough
begins to clear the road of snow. The speed, v km/h, of the snow plough is inversely
A
proportional to the depth of snow. (This means v  where A is a constant.)
h
Let x km be the distance the snow plough has cleared and let t be the time in hours from the
beginning of the snowfall. Let t = T correspond to 6 am.
dx k
i.
Explain carefully why, for t  T,
 , where k is a constant.
dt t
ii.
In the period from 6 am to 8 am the snow plough clears 1 km of road, but it takes a
further 3·5 hours to clear the next kilometre. At what time did it begin snowing?¤
« i) Proof ii) 3:20 am »
2U01-7c
t2
A particle moves in a straight line so that its displacement, in metres, is given by x 
t2
where t is measured in seconds.
i.
What is the displacement when t = 0?
4
ii.
Show that x  1 
. Hence find the expressions for the velocity and the
t2
acceleration in terms of t.
iii.
Is the particle ever at rest? Give reasons for your answer.
iv.
What is the limiting velocity of the particle as t increases indefinitely?¤
4
8
« i) x = –1 ii) v 
,a 
2
(t  2)
(t  2) 3
iii) No. The velocity is always greater than zero iv) 0 ms–1 »
2U01-8a
In November 1923, 18 koalas were introduced on Kangaroo Island. By November 1993, the
number of koalas had increased to 5000. Assume that the number N of koalas is increasing
exponentially and satisfies an equation of the form N = N0ekt, where N0 and k are constants
and t is measured in years from November 1923. Find the values of N0 and k, and predict the
number of koalas that will be present on Kangaroo Island in November 2001.¤
« N0 = 18, k = 00803163, 9512 koalas »
2U01-9b
When a valve is released, a chemical flows into a large tank that is initially empty. The
dV
 2e t  2e t where t is
volume, V litres, of chemical in the tank increases at the rate
dt
measured in hours from the time the valve is released.
i.
At what rate does the chemical initially enter the tank?
ii.
Use integration to find an expression for V in terms of t.
iii.
Show that 2e2t – 3et – 2 = 0 when V = 3.
iv.
Find t, to the nearest minute, when V = 3.¤
« i) 4 L/hr ii) V = 2et – 2e–t iii) Proof iv) 42 minutes »
2U01-10b
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
323
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
F Farmhouse
Claire’s

path
250 m
Front
gate G
D Bus depot
P
NOT TO
SCALE
2000 m
66)!
67)!
68)!
The diagram shows a farmhouse F that is located 250 m from a straight section of road. The
road begins at the bus depot D, which is situated 2000 m from the front gate G of the
farmhouse. The school bus leaves the depot at 8 am and travels along the road at a speed of
15 ms–1. Claire lives in the farmhouse, and she can run across the open paddock between the
house and the road at a speed of 4 ms–1. The bus will stop for Claire anywhere on the road,
but will not wait for her. Assume that Claire catches the bus at the point P on the road where
 GFP = .
i.
Find two expressions in terms of , one expression for the time taken for the bus to
travel from D to P and the other expression for the time taken by Claire to run from
F to P.
ii.
What is the latest time that Claire can leave home in order to catch the bus?¤
5(8  tan )
25 sec 
minutes, T2 
minutes
« i) T1 
18
24
ii) 1 minute 13 seconds past 8 am »
2U02-7b
A cooler, which is initially full, is drained so that at time t seconds the volume of water V, in
t
litres, is given by V  25(1  ) 2 for 0  t  60 .
60
i.
How much water was initially in the cooler?
ii.
After how many seconds was the cooler one-quarter full?
iii.
At what rate was the water draining out when the cooler was one-quarter full?¤
5
L/s »
« i) 25 L ii) 30 seconds iii)
12
2U02-8a
A drug is used to control a medical condition. It is known that the quantity Q of drug
remaining in the body after t hours satisfies an equation of the form Q = Q0e–kt where
Q0 and k are constants. The initial dose is 6 milligrams and after 15 hours the amount
remaining is the body is half the initial dose.
i.
Find the value of Q0 and k.
ii.
When will one-eighth of the initial dose remain?¤
« i) Q0 = 6, k = 0046 (to 3 d.p) ii) 45 hours »
2U02-8b
A particle moves in a straight line. At time t seconds, its distance x metres from a fixed
point O on the line is given by x = sin 2t + 3.
i.
Sketch the graph of x as a function of t for 0  t  2.
ii.
Using your graph, or otherwise, find the times when the particle is at rest, and the
position of the particle at those times.
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
324
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
iii.
Describe the motion completely.¤
y
4
3
2
  3  5 3 7 2
69)!
2
4
4
2 4
3
5
7
( x  2), t 
( x  4), t 
( x  4) ii) Every  seconds, the particle
4
4
4
4
oscillates on a straight line between x = 2 and x = 4 around the centre of motion (x = 3) »
2U02-9c
ii) t 

4
« i)
t
( x  4) , t 
v(m/s)
50
Car
Jet
O
5
NOT TO
SCALE
t(seconds)
A car and a jet race one another from rest down a runway. The car increases its speed v1 at a
constant rate, while the speed of the jet is given by v2 = 2t2. After 5 seconds the car and the
jet have the same speed of 50 m/s, as shown on the graph.
i.
Find an equation for the speed v1 of the car in terms of t.
ii.
How far behind the car is the jet after 5 seconds?
iii.
After how many seconds does the jet catch up with the car?¤
2
1
« i) v1 = 10t ii) 41 m iii) 7 seconds »
3
2
70)! 2U03-6c
A farmer accidentally spread a dangerous chemical on a paddock. The concentration of the
chemical in the soil was initially measured at 5 kg/ha. One year later the concentration was
found to be 28 kg/ha. It is known that the concentration, C, is given by C = C0e–kt, where
C0 and k are constants, and t is measured in years.
i.
Evaluate C0 and k.
ii.
It is safe to use the paddock when the concentration is below 02 kg/ha. How long
must the farmer wait after the accident before the paddock can be used? Give your
answer in years, correct to one decimal place.¤
« i) C0 = 5, k = 05798 (to 4 d.p) ii) 56 years »
71)! 2U03-7b
The velocity of a particle is given by v = 2 – 4cos t for 0  t  2, where v is measured in
metres per second and t is measured in seconds.
i.
At what times during this period is the particle at rest?
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
325
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
ii.
iii.
iv.
What is the maximum velocity of the particle during this period?
Sketch the graph of v as a function of t for 0  t  2.
Calculate the total distance travelled by the particle between t = 0 and t = .¤
v
6
4
2
« i) t 
 5
3
,
3
seconds ii) 6 ms–1 iii)

–2
5
3

3
4 3
72)!
« i) 4 metres to the right of O ii)
iv)
2π
metres »
3
x = 1 +3cos2t
1
O
–2
x = 1 (Centre of motion)


4
2
3
4

t
and x = 2 iv) When t 
74)!
t
2U04-5b
A particle moves along a straight line so that its displacement, x metres, from a fixed point O
is given by x = 1 + 3 cos 2t, where t is measured in seconds.
i. What is the initial displacement of the particle?
ii. Sketch the graph of x as a function of t for 0 t  .
iii. Hence, or otherwise, find when AND where the particle first comes to rest after t = 0.
iv. Find a time when the particle reaches its maximum speed. What is this speed?¤
x
4
73)!
2
iii) When t 

4

2
, the speed is 6 m/s »
2U04-7b
At the beginning of 1991 Australia’s population was 17 million. At the beginning of 2004 the
population was 20 million. Assume that the population P is increasing exponentially and
satisfies an equation of the form P  Ae kt , where A and k are constants, and t is measured in
years from the beginning of 1991.
dP
 kP.
i. Show that P  Ae kt satisfies
dt
ii. What is the value of A?
iii. Find the value of k.
iv. Predict the year during which Australia’s population will reach 30 million.¤
« i) Proof ii) 17 000 000 iii) k = 0∙0125 (to 4 dp) iv) 2036 »
2U04-9b
A particle moves along the x-axis. Initially it is at rest at the origin. The graph shows the
acceleration, a, of the particle as a function of time t for 0  t  5.
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
326
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
a
3
(1, 3)
1 2
–3
3
(3, –3)
4 5
t
(5, –3)
i.
ii.
75)!
76)!
Write down the time at which the velocity of the particle is a maximum.
At what time during the interval 0  t  5 is the particle furthest from the origin? Give
brief reasons for your answer.¤
« i) t = 2 ii) t = 4 »
2U05-6b
A tank initially holds 3600 litres of water. The water drains from the bottom of the tank. The
tank takes 60 minutes to empty. A mathematical model predicts that the volume, V litres, of
t
water that will remain in the tank after t minutes is given by V  3600 (1  ) 2 , where
60
0 ≤ t ≤ 60.
i.
What volume does the model predict will remain after ten minutes?
ii.
At what rate does the model predict that the water will drain from the tank after
twenty minutes?
iii.
At what time does the model predict that the water will drain from the tank at its
fastest rate?¤
« i) 2500 L ii) 80 L/min iii) t = 0 »
2U05-7b
dx
dt
(1, 2)
2
O
–2
The graph shows the velocity,
2
6
(3, –2)
t
(5, –2)
dx
, of a particle as a function of time. Initially the particle is
dt
at the origin.
i.
At what time is the displacement, x, from the origin a maximum?
ii.
At what time does the particle return to the origin? Justify your answer.
d 2x
iii.
Draw a sketch of the acceleration, 2 , as a function of time for 0 ≤ t ≤ 6.¤
dt
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
327
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
d2y
dt 2
0
« i) t = 2 ii) t = 4 iii)
77)!
1
2
3
4
5
6
t
»
2U05-9a
A particle is initially at rest at the origin. Its acceleration as a function of time, t, is given by
x  4 sin 2t .
i.
Show that the velocity of the particle is given by x  2  2 cos 2t .
ii.
Sketch the graph of the velocity for 0 ≤ t ≤ 2π AND determine the time at which the
particle first comes to rest after t = 0.
iii.
Find the distance travelled by the particle between t = 0 and the time at which the
particle first comes to rest after t = 0.¤

x
.
x  2  2cos2t
4
2
0
78)!
79)!

4

2
3  5 3
4
4 2
7 2 t
4
t = π iii) 2π units »
« i) Proof ii)
2U06-6b
A rare species of bird lives only on a remote island. A mathematical model predicts that the
bird population, P, is given by
P = 150 + 300e–005t
Where t is the number of years after observations began.
i.
According to the model, how many birds were there when observations began?
ii.
According to the model, what will be the rate of change in the bird population ten
years after observations began?
iii.
What does the model predict will be the limiting value of the bird population?
iv.
The species will become eligible for inclusion in the endangered species list when
the population falls below 200. When does the model predict that this will occur?¤
« i) 450 ii) 9 fewer birds each year iii) 150 iv) The 36th year »
2U06-8a
A particle is moving in a straight line. Its displacement, x metres, from the origin, O, at time t
7
seconds, where t  0, is given by x  1 
.
t4
i.
Find the initial displacement of the particle.
ii.
Find the velocity of the particle as it passes through the origin.
iii.
Show that the acceleration of the particle is always negative.
iv.
Sketch the graph of the displacement of the particle as a function of time.¤
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
328
M_BANK\YR12-2U\APPLICATIONSOFCALCULUS.HSC
x
1
x  1 7
t 4
0
3
t
3

4
3
1
m from O ii) ms -1 iii) Proof iv)
4
7
2U06-9b
« i) 
80)!
During a storm, water flows into a 7000 litre tank at a rate of
»
dv
litres per minute, where
dt
dv
 120  26 t  t 2 and t is the time in minutes since the storm began.
dt
i.
At what times is the tank filling at twice the initial rate?
ii.
Find the volume of water that has flowed into the tank since the start of the storm as
a function of t.
iii.
Initially, the tank contains 1500 litres of water. When the storm finishes, 30 minutes
after it began, the tank is overflowing. How many litres of water have been lost?¤
1
« i) After 6 minutes and after 20 minutes ii) V = 120t + 13t2 – t 3 + c iii) 800 L »
3
¤©BOARD OF STUDIES NSW 1984 - 2006
©EDUDATA SOFTWARE PTY LTD: DATA VER5.0 2006
329