# 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.

2.
Which of the following equations represents exponential growth?
 a. $y$ = $r$(1 + $r$) b. $y$ = $r$(1 + C) c. $y$ = C$r$ 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.

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.

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

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

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.

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.

7.

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.

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.]

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.