Calculus 12
1.
Worksheet #1
Accumulators
Chapter 7
The rate at which water is leaking from a tank (in L/h) is given by the equation:
L(t )  4t 2  20t  5 where t is the time in hours.

4
95.52 L
a)
Use a midpoint Riemann sum with 5 equal intervals to estimate
b)
What are the units to
c)
If the tank started with 103.5 L of water, how long will it take for the leak to empty the
tank?
4.5 h
What was the maximum rate of water leak (include units) and when did it occur?
d)

4
0
0
L(t ) dt
L(t ) dt and what does this integral represent? total amount leaked
L( 2.5h ) = 30 L/h
2.
The amount of current I flowing through a wire is a measure of the rate of flow of charge,
measured in Coulombs/Second. The current in a particular wire is show on the graph below
I


t


75 Coulombs

a)
Estimate the amount of charge that has flowed through the wire in the first 2 seconds.
b)
Estimate the amount of charge that has flowed through the wire in the first 4 seconds.
c)
What is the average flow of current in the first 3 seconds?
100 Coulombs
32 Coulombs / s
3.
Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per
minute. The traffic flow at a particular intersection is modeled by the function F defined by
t
F (t )  82  4 Sin   for 0  t  30,
2
where F(t) is measured in cars per minute and t is measured in minutes.
a)
To the nearest whole number, how many cars pass through the intersection over the
30-minute time period?
2474 Cars
b)
Is the traffic flow increasing or decreasing at t=7? Justify your answer.
F’(7)=-1.8
decreasing
What is the average value of the traffic flow over the time interval 10  t  15 ?
Indicate units of measure.
81.8992 Cars/min
increasing
What is the average rate of change
of the traffic flow over the time interval 10  t  15 ?
c)
d)
Indicate units of measure.
1.5175 Cars/min2
increasing
Calculus 12
1.
Worksheet #2
Accumulators
Chapter 7
Oil is leaking out of a tanker at a rate of R(t )  2000 e 0.2 t liters per hour. How much oil has
leaked out of the tanker after 10 hours?
8646.6472 L
2.
The rate at which people enter an amusement park on a given day is given by the function:
15600
E (t )  2
t  24t  160
The rate at which people leave the same amusement park is given by the function:
9890
L(t )  2
t  38t  370
Both E(t) and L(t) are measured in people per hour and t is measured from midnight.
The park is open from 9  t  23 . At t=9 there is no one in the park.
a)
b)
c)
d)
How many people have entered the park by t=17? 6004
1271
How many people enter the park between t=17 and t=23?
Ticket prices are $15 until 5PM (t=17) and then only $11 until the close of the park. How
much revenue does the park make in one day?
$ 104 041
When is the number of people in the park a maximum?
increasing
t=15.8000
3.
A long lake is used as a reservoir to supply a hydro dam. The lake is approximately the shape of a
rectangular prism (box) with a length of 80 km, a width of 2 km and a depth of 500 m. On
January 1st, the amount of water in the reservoir is 80 billion m3.
The flow of water into the lake is modeled by the function
90
E (t ) 
, 0  t  365
 2 
2  Cos 
t
 365 
where E (t ) is measured in millions of m3/day, and t is measured in days.
The flow of water out of the lake is used for electrical supply and is higher during the
colder Winter months. It is modeled by the function
146
L(t ) 
, 0  t  365
 2 
3  Cos 
t
 365 
where L (t ) is measured in millions of m3/day, and t is measured in days.
over,
a)
On the same grid below sketch the curves E(t) and L(t).
y
 F








t
x


























b)
What is the total amount of water added into to the reservoir for the entire year?
Express your answer in millions of m3
18 969.956 mill m3
c)
What is the total amount of water leaving the reservoir for the entire year?
Express your answer in millions of m3
18 840.86 mill m3
Describe the long term status of the reservoir based on your answers to “b” and “c”.
d)
e)
Sketch the net water
flow for the year on
the grid on the right
















y
growing slightly
F
t



        
x



        
f)
What was the change in depth of the lake 1 year later?
Express your answer in metres
+0.7819 m
g)
On what day of the year is the lake at its highest level?
250th
What is the depth of the lake on this day (in metres)?
504.3940 m
On what day of the year is the lake at its lowest level?
115th
What is the depth of the lake on this day (in metres)?
496.3879 m
On what day of the year is the lake level rising the fastest?
182nd
What is the rate of depth increase on this day (in cm/hour)
h)
i)
j)
k)
l)
+0.4427
cm/h
Calculus 12
Worksheet #3
Accumulators
1.
Find the particular solution to y '  3 y  6 , with y (0)  7
2.
If y '  Sin  x 2  , and y (3)  7 , find y (5)
3.
The function f (t ) is defined in piece by:
Chapter 9
y  9 e3 x  2
6.7544
 4  t 2 ,  2  t  2
f (t )  
 1  t  3 , 2  t  5
x
Let g be defined by g ( x)   f (t ) dt
0
a)
b)
Sketch f (t )
Find an exact value for g(3).
  12
y
y
c)


Find all values of x on the open interval
(2,5) at which g has a relative maximum.
Justify your answer.
x2
x x d)





e)


4.
Write an equation for the tangent line to the
graph of g at x  3 .
y  (  12 )  1( x  3)
or, y   x  52  
Find the x-coordinate for each point of
inflection of the graph of g on the open
interval (2,5) . Justify your answer.
x 0 , x 3
The rate at which water flows into a tank, in liters per minute, is given by a differentiable function
R of time t. The table below shows the rate as measured every 5 minutes over a 60 min period.
t
0
5
10
15
20
25
30
35
40
45
50
55
60
R (t )
11.3
14.6
17.2
19.0
20.0
20.2
19.7
18.4
16.2
13.4
9.7
5.2
0
a)
Use a midpoint Riemann Sum with 6 equal
subdivisions to approximate

60
0
R(t ) dt .
Using correct units, explain the meaning of
your answer in terms of water flow.
908 L = total water flowed
b)
Is there a time 0  t  60 such that
R '(t )  0 . Justify your answer.
Yes, between t=20 and t=30
c)
Use your answer from part “a” to
approximate the average water flow over
the 60 min period.
15.1333 L/min
Calculus 12
1.
Worksheet #4
Chapter 9
Forty kilograms of salt is dissolved in a container holding 800 L of water. A salt solution with a
concentration of 0.200 kg/L of salt is added to the tank at a rate of 12 L/min. The tank is well
mixed and the mixed solution flows out of the tank also at a rate of 12 L/min.
a)
Write a differential equation describing the kilograms of salt within the tank
ds
12
 2.4  800
s
dt
b)
Solve the resulting differential equation
s  160  120e
2.
Mixing Questions
12 t
 800
c)
When will there be 120 kg of salt in the tank?
t  73.2408 min
d)
What is the concentration in the tank after 2 hours? (kg/L)
 s  0.1752 kg L
e)
At what rate is salt leaving the tank after 1 hour?
ds
 1.6682 kg min
dt
(in kg/min)
The amount of heat energy in a litre of water is given by the formula Eh  4180T . An 8000L hot
tub is full of overheated water with a temperature of 60oC. Cooler water at a constant temperature
of 15oC is continuously added to the hot tub at a rate of 10 kg/min. Well mixed water is then
allowed to overflow out of the hot tub into the drain.
a)
Write a differential equation describing the amount of heat energy within the tank
dE
10
 627000  8000
E
dt
b)
Solve the resulting differential equation
E  501600000  1504800000 e
c)
10 t
 8000
When will the temperature of the hot tub reach a safe 40oC?
t  470.2293 min
d)
What is the temperature of the hot tub if the cool water flows for 5 hours?
T  45.9280 oC
3.
A completely full 2.00 L beaker of 0.75 M [Na+] solution is mixed continuously while a solution
of 0.45 M [Na+] is added at a rate of 0.01 L/min.
a)
Let y(t)= the amount of mol of Na+ in the water and write a differential equation for y(t)
dy
 0.045  0.005 y
dt
b)
Solve for y(t) by separating the differential equation.
y  0.9  0.6 e 0.005t
c)
When will there be 1.0 mol of Na+ in the solution
t  358.3520 min
4.
A full 5.0 L container of water is colored blue by the addition of 30 drops of blue food coloring.
A second much larger 200 L container has been colored red by the addition of 800 drops of red
food coloring. The red colored water is then added to the 5.0 L container of blue water at a flow
rate of 0.1 L/min. The solution is evenly mixed and the excess liquid spills away.
When will the 5.0 L container have the most purple looking water? i.e. [Red]=[Blue]
t  45.8145 min
5.
A 2 L beaker is full to the top with 4.0 M [ H 2 SO4 ] . A second solution of 3.0 M [ NaOH ] is added
at a rate of 10 mL/s while the well mixed solution is drained away. How long will it take to reach
a neutral solution with a pH of 7.0?
*
*
Remember the H 2 SO4 yields two H ions but the NaOH only yields one OH ion.
A neutral solution has equal amounts of H and OH.
t  259.8566 s