IB Math Studies 2
Applications of Growth and Decay and Sample IB test questions
1. The following diagram shows part of the graph of an exponential function f(x) = a–x, where x.
(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 D.
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.
(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). ______
4. 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.
(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.
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. A function is represented by the equation f (x) = 3(2)x + 1.
The table of values of f (x), for – 3 x 2, is given below.
x / –3 / –2 / –1 / 0 / 1 / 2f (x) / 1.375 / 1.75 / a / 4 / 7 / b
(a) 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.______
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.
7 July
(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) (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. ______
(b) Find how many years it will take for the value of the car to be less than 8000 USD. ______
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 / 16No. 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 where t is the time (in months) after the beginning of the colony .
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). ______