IB Math Studies 2
Applications of Growth and Decay and Sample IB test questions
APPLYING EXPONENTIAL EQUATIONS TO SOLVE PROBLEMS
f(x) = kaλx + c where λ is the rate of growth or decay and x is the time
f(x) = eλx + c where λ is the rate of growth or decay and x is the time
Appreciation: V = a(1 + r)t , a is initial value, r is the rate as a decimal, t is time
Depreciation: V = a(1 – r)t , a is initial value, r is the rate as a decimal, t is time
1. The following diagram shows part of the graph of an exponential function f(x) = a–x, where x
.
y
f(x)
P
x
0
(a)
What is the range of f ?
_________________________
(b)
Write down the coordinates of the point P.
_________________________
(c)
What happens to the values of f(x) as elements in its domain increase in value?
2. The figure below shows the graphs of the functions y = x2 and y = 2x for values of x between –2 and 5. The
points of intersection of the two curves are labelled as B, C and
20
y
(a)
Write down the coordinates of the point A.
(b)
Write down the coordinates of the points B
and C.
(c)
Find the x-coordinate of the point D.
C
15
10
5
D
D. –2
–1
B
A
1
2
3
4
5
(d) Write down, using interval notation, all values
x
x
2
of x for which 2 ≤ x .
IB Math Studies 2
Applications of Growth and Decay and Sample IB test questions
3. The figure below shows the graphs of the functions f (x) = 2x + 0.5 and g (x) = 4 − x2 for values of x
between –3 and 3.
y
f(x)
6
B
3
A
–3
–2
–1
0
–3
3 x
2
g(x)
(a)
Write down the coordinates of the points A and B. ______________________
(b)
Write down the set of values of x for which f (x)< g (x). ______________________
The following graph shows the temperature in degrees Celsius of Robert’s cup of coffee, t minutes after
pouring it out. The equation of the cooling graph is f (t) =16 + 74 × 2.8−0.2t where f (t) is the temperature and
t is the time in minutes after pouring the coffee out.
100
Temperature (°C )
4.
1
80
60
40
20
0
0 1
2
3 4
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Time (minutes)
(a)
Find the initial temperature of the coffee.
_________________
(b)
Write down the equation of the horizontal asymptote.
_________________
(c)
Find the room temperature.
_________________
(d)
Find the temperature of the coffee after 10 minutes.
_________________
If the coffee is not hot enough it is reheated in a microwave oven. The liquid increases in temperature
according to the formula:
T = A × 21.5t
where T is the final temperature of the liquid, A is the initial temperature of coffee in the microwave and t is
the time in minutes after switching the microwave on.
IB Math Studies 2
Applications of Growth and Decay and Sample IB test questions
Number 4 continued…
(e)
Find the temperature of Robert’s coffee after being heated in the microwave for 30 seconds after it
has reached the temperature in part (d).
_____________________
(f)
Calculate the length of time it would take a similar cup of coffee, initially at 20 , to be heated in the
microwave to reach 100 .
______________________
5.
x
A function is represented by the equation f (x) = 3(2) + 1.
The table of values of f (x), for – 3
(a)
x
2, is given below.
x
–3
–2
–1
0
1
2
f (x)
1.375
1.75
a
4
7
b
Calculate the values for a and b.
_________________
(2)
(b)
On graph paper, draw the graph of f (x) , for – 3
x
2, taking 1 cm to represent 1 unit on both axes.
(4)
The domain of the function f (x) is the real numbers,
(c)
.
Write down the range of f (x).
_________________
(2)
(d)
Using your graph, or otherwise, find the approximate value for x when f (x) = 10.
_________________
(2)
6.
In an experiment researchers found that a specific culture of bacteria increases in number according to the
formula
N = 150 × 2t,
where N is the number of bacteria present and t is the number of hours since the experiment began.
Use this formula to calculate
(a)
the number of bacteria present at the start of the experiment; ___________________
(b)
the number of bacteria present after 3 hours;
(c)
the number of hours it would take for the number of bacteria to reach 19 200._______
_____________________
IB Math Studies 2
Applications of Growth and Decay and Sample IB test questions
7.
The area, A m2, of a fast growing plant is measured at noon (12:00) each day. On 7 July the area was 100
m2. Every day the plant grew by 7.5%. The formula for A is given by
A = 100 (1.075)t
where t is the number of days after 7 July. (on 7 July, t = 0)
The graph of A = 100(1.075)t is shown below.
A
400
300
200
100
–6
–4
–2 0
7 July
2
4
6
8
10
12
14
16
18
t
(a)
What does the graph represent when t is negative? ____________________
(b)
Use the graph to find the value of t when A = 178. ______________________
(c)
Calculate the area covered by the plant at noon on 28 July. _____________________
8. The value of a car decreases each year. This value can be calculated using the function
v = 32 000rt, t
,
where v is the value of the car in USD, t is the number of years after it was first bought and r is a constant.
(a)
(b)
(i)
Write down the value of the car when it was first bought. _____________
(ii)
One year later the value of the car was 27 200 USD. Find the value of r. _______________
Find how many years it will take for the value of the car to be less than 8000 USD. ______________
IB Math Studies 2
Applications of Growth and Decay and Sample IB test questions
9. In an experiment it is found that a culture of bacteria triples in number every four hours. There are 200
bacteria at the start of the experiment.
Hours
0
4
8
12
16
No. of bacteria
200
600
a
5400
16200
(a)
Find the value of a. _______________
(b)
Calculate how many bacteria there will be after one day. _____________
(c)
Find how long it will take for there to be two million bacteria. ____________
10. The number of ants, N (in thousands), in a colony is given by
the beginning of the colony
where t is the time (in months) after
.
a) Calculate the initial number of ants at the start of the colony _________________
b) Calculate the number of ants present after 2 months ___________________
c) Find the time taken for the colony to reach 20,000 ants ____________________
d) Determine the equation of the horizontal asymptote of N(t) _________________
e) According to the function N(t), what is the largest number of ants the colony will never reach? _____________
11. The population affected by a virus grows at a rate of 20% per day. Initially there are 10 people affected.
a) Find the number of people affected after 1 day. _______________
b) Find the number of people affected after 1 week. Give your answer correct to the nearest whole number.
_______________
c) Let the number of people affected after t days be given by
.
State the value of:
i) N _________________________ ii) a __________________________
d) Using the graphical display calculator sketch the graph of f(t) for
showing clearly the value of the y-
intercept.
e) Write down the range of f(t) for the given domain. Give your answer correct to the nearest whole number.
__________________
f) Write down the equation of the horizontal asymptote of f(t). ___________________