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Exponential Growth Worksheet

Exponential Growth Worksheet
  • Page 1
 1.  
If a quantity increases by the same percent r in each unit of time t, then the quantity is ____________.
a.
growing exponentially
b.
decreasing exponentially
c.
constant


Solution:

If a quantity is increasing by the same percent r in each unit of time t, then the quantity is growing exponentially.


Correct answer : (1)
 2.  
Which of the following equations represents exponential growth?
a.
y = r(1 + r)
b.
y = r(1 + C)
c.
y = Cr
d.
y = C(1 + r)t


Solution:

Exponential growth can be modeled by the equation y = C (1 + r)t, where C is the initial amount, r is the growth rate and t is the time.


Correct answer : (4)
 3.  
The expression (1 + r) is called ______ in the equation y = C(1 + r)t.
a.
decay factor
b.
growth factor
c.
decay and growth factors
d.
exponent


Solution:

The expression (1 + r), in the equation y = C(1 + r)t is called growth factor.


Correct answer : (2)
 4.  
The average length of a person's hair at birth is 0.36 inches. The length of the hair increases by about 10% each day during the first six weeks. Choose a model that represents the average length of the hair during the first six weeks.
a.
y = 0.36(1.1)t
b.
y = -0.36(1.1)t
c.
y = 1.1(0.36)t
d.
None of the above


Solution:

Let y be the length of the hair during the first six weeks and t be the number of days.

y = C(1 + r)t
[Write exponential growth model.]

= 0.36(1 + 0.10)t
[Substitute C = 0.36 and r = 0.10.]

= 0.36(1.1)t
[Add.]

The model for the length of the hair in first six weeks is y = 0.36(1.1)t.


Correct answer : (1)
 5.  
A bank pays 4% interest compounded yearly on a deposit of $900. What will be the balance in the account after 7 years?
a.
$1288
b.
$1088
c.
$2376
d.
$1188


Solution:

The exponential growth model is given by the equation, y = P(1 + r)t, where P is the initial amount, r is the growth rate and t is the number of years.

= 900(1 + 0.04)7
Balance after 7 years
[Substitute P = 900, t = 7 and r = 0.04.]

= 900(1.04)7 = 900 x 1.32 = 1188
[Simplify.]

The account balance after 7 years will be about $1188.


Correct answer : (4)
 6.  
There are 20 bears in a zoo. What will be their population after 3 years, if the population doubles each year?
a.
160 bears
b.
260 bears
c.
60 bears
d.
210 bears


Solution:

The exponential growth model is given by the equation, y = C(1 + r)t, where C is the initial number, (1 + r) is the growth factor and t is the number of years.

Population after 3 years = 20(2)3
[Substitute C = 20, 1 + r = 2 and t = 3.]

= 160
[Simplify.]

There will be 160 bears after 3 years.


Correct answer : (1)
 7.  
The cost of a movie ticket is $8, which increases by 3% each year. Find the cost of the ticket after 8 years.
a.
$15.00
b.
$10.00
c.
$20.00
d.
$64


Solution:

The related equation is P = A (1 + r)t, where P is the price, t is the time (in years) and A is the initial cost.

P = 8 (1 + 0.03)8
[Substitute A = 8, r = 0.03 and t = 8.]

= 8(1.03)8 = 8 x 1.27 = 10.16
[Simplify.]

The ticket costs about $10.00 after 8 years.


Correct answer : (2)
 8.  
Rents in a particular area are increasing by 3% every year. Predict what the rent of the apartment would be after 2 years, if its rent is $400 per month now.
a.
$424.36
b.
$425.36
c.
$434.36
d.
$435.36


Solution:

The related model is P = A(1 + r)t where, P is the price, t is the time in years and A is the initial rent.

P = 400(1 + 0.03)2
[Substitute A = 400, r = 0.03 and t = 2.]

= 400(1.03)2 = 424.36
[Simplify.]

The rent of the apartment after 2 years will be $424.36.


Correct answer : (1)
 9.  
What is the percent increase in the growth of a tree, if the initial height of the tree is one foot and the height after one year is 1.40 feet?
a.
30%
b.
50%
c.
60%
d.
40%


Solution:

Consider the equation y = C(1 + r)t
[Exponential growth model.]

From the above equation, the growth factor is given by (1 + r), where r is the percent increase in the height of the tree.

1.40 can be written as (1 + 0.40)

So, the percent increase is 0.40 = 40%
[Compare with the exponential growth equation.]


Correct answer : (4)
 10.  
The population of the United States was about 250 million in 2003, and is growing exponentially at a rate of about 0.7% per year. What will be its population in the year 2013?
a.
250 million
b.
268.1 million
c.
248.1 million
d.
230 million


Solution:

From the data, the time t is 10 years, since it calculated from the year 2003 to year 2013.

The percent increase, r = 0.7% = 0.007

The initial population, C = 250 million

y = C(1 + r)t
[Write exponential growth model.]

= 250(1 + 0.007)10
[Substitute C = 250, r = 0.007 and t = 10.]

= 250(1.007)10 = 268.1
[Simplify.]

The population in the year 2013 will be 268.1 million.


Correct answer : (2)

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