Revision - Exponential functions
1.
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.
Working:
Answers:
(a) …………………………………………..
(b) …………………………………………..
(c) …………………………………………..
(Total 4 marks)
Revision Exp Functions IB10
MaWa 2013
2.
The graph below shows the curve y = k(2x) + c, where k and c are constants.
y
10
–6
–4
–2
0
2
4
x
–10
Find the values of c and k.
Working:
Answers:
.......…………………………………………..
..........................................................................
(Total 4 marks)
Revision Exp Functions IB10
MaWa 2013
3.
The diagram shows a chain hanging between two hooks A and B.
The points A and B are at equal heights above the ground. P is the lowest point on the
chain. The ground is represented by the x-axis. The x-coordinate of A is –2 and the xcoordinate of B is 2. Point P is on the y-axis. The shape of the chain is given by y = 2x +
2–x where –2 ≤ x ≤ 2.
A
B
y
Chain
P
Ground
–2
0
2
x
(a)
Calculate the height of the point P.
(b)
Find the range of y. Write your answer as an interval or using inequality symbols.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 8 marks)
Revision Exp Functions IB10
MaWa 2013
4.
The number of bacteria (y) present at any time is given by the formula:
y = 15 000e–025t, where t is the time in seconds and e = 2.72 correct to 3 s.f.
(a)
Calculate the values of a, b and c to the nearest hundred in the table below:
Time in seconds (t)
0
1
2
3
Amount of bacteria (y) a 11700 9100 7100
(nearest hundred)
4
b
5
6
7
4300 3300 2600
8
c
(3)
(b)
On graph paper using 1 cm for each second on the horizontal axis and 1 cm for
each thousand on the vertical axis, draw and label the graph representing this
information.
(5)
(c)
Using your graph, answer the following questions:
(i)
After how many seconds will there be 5000 bacteria? Give your answer
correct to the nearest tenth of a second.
(ii)
How many bacteria will there be after 6.8 seconds? Give your answer correct
to the nearest hundred bacteria.
(iii)
Will there be a time when there are no bacteria left? Explain your answer.
(6)
(Total 14 marks)
Revision Exp Functions IB10
MaWa 2013
5.
The diagram below shows a part of the graph of y = ax. The graph crosses the y-axis at the
point P. The point Q (4, 16) is on the graph.
y
Q
(4, 16)
P
Diagram not to scale
x
O
Find
(a)
the coordinates of the point P;
(b)
the value of a.
Working:
Answers:
(a) ..................................................................
(b) ……………………………………..........
(Total 8 marks)
Revision Exp Functions IB10
MaWa 2013
6.
The equation M = 90 × 2–t/20 gives the amount, in grams, of radioactive material held in a
laboratory over t years.
(a)
What was the original mass of the radioactive material?
The table below lists some values for M.
t
60
80
100
M
11.25
v
2.8125
(b)
Find the value of v.
(c)
Calculate the number of years it would take for the radioactive material to have a
mass of 45 grams.
Working:
Answers:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(Total 8 marks)
Revision Exp Functions IB10
MaWa 2013
7.
The value of a car decreases each year. This value can be calculated using the function
v = 32 000rt, t
0, 0
r
1,
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.
Working:
Answers:
(a)
(i)..........................................
(ii).........................................
(b) .................................................
(Total 6 marks)
Revision Exp Functions IB10
MaWa 2013
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 )
8.
(a)
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)
Find the initial temperature of the coffee.
(1)
(b)
Write down the equation of the horizontal asymptote.
(1)
(c)
Find the room temperature.
(1)
(d)
Find the temperature of the coffee after 10 minutes.
(1)
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.
(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).
(3)
(f)
Calculate the length of time it would take a similar cup of coffee, initially at 20 C,
to be heated in the microwave to reach 100 C.
(4)
(Total 11 marks)
Revision Exp Functions IB10
MaWa 2013
Answers to the revision exercise on Exponential Functions
1.
(a)
N = 150×20 = 150
(A1) (C1)
(b)
N = 150×23 = 1200
(A1) (C1)
(c)
19200 = 150×2t
128 = 2t
7=t
(M1)
(A1) (C2)
[4]
2.
c = – 10 (asymptote of graph)
0 = k(21) – 10 2k = 10
k=5
(M1)(A1)
(M1)
(A1)
OR
k + c = –5
2k + c = 0
Therefore, k = 5
c = – 10
(M1)
(M1)
(A1)
(A1)
[4]
3.
(a)
Point P is at x = 0. Find y(0).
y(0) = 20 + 2–0
=1+1=2
(b)
(0, 2) is the lowest point.
Highest points are at x = 2 or x = –2.
At x = 2 or –2, y = 22 + 2–2 = 4 14 or 4.25
So the range is [2, 4 14 ] or 2
(M1)
(A1)
(A1) (C3)
y
4.25.
(M1)(A1)
(A1)(A1)(A1) (C5)
Note: Award (A1) for each inequality (or bracket).
If both inequalities are strict (or parentheses)
award (A0)(A1)(ft).
Award (A1) for both numbers in order.
[8]
4.
(a)
a = 15000
b = 5500
c = 2000
(A1)
(A1)
(A1)
Revision Exp Functions IB10
MaWa 2013
3
(b)
15 000
14 000
13 000
12 000
11 000
10 000
number
of
bacteria
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
1
2
5
3
6
4
time (in seconds)
7
8
9
5
Note: Award (A1) for axes correctly labelled, (A1) for
correct scales, (A1) for smooth curve, (A2) for all points
correctly plotted, (A1) for at least 4 points correct.
(c)
(i)
4.4 secs
(M1)(A1)
Note: Award (M1)(A1)(ft) from graph (see (b)) or (A1) if
correct and no line seen.
(ii)
2700 bacteria (±200 bacteria)
(M1)(A1)
Note: Award (M1)(A1)(ft) from graph (see (b)) or (A1) if
correct and no line seen.
(iii)
No ― theoretically, the curve never touches the horizontal axis
(or any answer to suggest that the horizontal axis is
an asymptote).
(A1)(R1)
Note: Award (A1)(R1) for any time over 39 seconds with
a reasonable explanation (when the number of bacteria is
less than one).
Award (A0)(R0) for a yes or no with no explanation. Do
not award (A1) if (R1) is not awarded.
6
[14]
5.
(a)
(0,1)
(b)
16 = a4
a=2
(A2)(A2) (C4)
(M2)
(A2) (C4)
[8]
Revision Exp Functions IB10
MaWa 2013
0
6.
(a)
M = 90 × 2 20 = 90 (grams)
Note: Award (M1) for t = 0.
80
20
(b)
t = 80…… M = 90 × 2
Therefore, v = 5.625 (grams) (5.63 3 s.f.) (accept either)
(c)
45 = 90 × 2 20
(M1)(A1) (C2)
(M1)
(A1) (C2)
t
(M1)
t
2 20
= 0.5
t = 20 years
(M1)
(A2) (C4)
[8]
7.
32 000r0 = 32 000
Award (M1) for putting t = 0.
(a)
(i)
(ii)
32 000r = 27 200
(M1)
r = 0.85
(b)
32 000
(M1)(A1) (C2)
(A1) (C2)
0.85t = 8000
(M1)
0.85t = 0.25
t = 8.53 (3 s.f.) (accept 9)
(A1)(ft) (C2)
[6]
8.
Unit penalty (UP) is applicable in part (i)(a)(c)(d)(e) and (f)
(a)
90 C
(A1)
(UP)
1
(b)
y = 16
(A1)
1
(A1)(ft)
(UP)
1
1
(c)
16 C (ft) from answer to part (b)
(d)
25.4 C
(A1)
(UP)
(e)
for seeing 20.75 or equivalent
(A1)
for multiplying their (d) by their 20.75
(M1)
42.8 C
(f)
for seeing 20
(A1)(ft)(G2)
(UP)
21.5t = 100
3
(A1)
for seeing a value of t between 1.54 and 1.56 inclusive
1.55 minutes or 92.9 seconds
(M1)(A1)
(A1)(G3)
(UP)
4
[11]
Revision Exp Functions IB10
MaWa 2013