PROBLEMS ON EXPONETIAL AND LINEAR GROWTH PRACTICE FOR SAT

Problem 1 :

If the initial population of rats was 20 and grew to 25 after the first year, which of the following function best models the population of rats P with respect to the number of years t if the population growth of rats is considered to be exponential ?

a)  P = 5t + 20    b)  P = 20 (1.25)^t      c)  P = 20(5)^t

d)  P = 5t² + 20

Solution :

Considering option A :

P = 5t + 20

P- population of rats, t - number of years.

Initial population = 20, population of rats after 1 year = 25, after 2 years = 30. So, option A is correct.

Problem 2 :

If the initial population of pandas was 100 and grew to 125 after the first year, which of the following function best models the population of pandas P with respect to the number of years t if the population growth of pandas is considered to be linear ?

a)  P = 25t + 100    b)  100 (1.25)^t        c)  P = 100(1.2)^t

d)  P = 20t² + 5t + 100

Solution :

Linear function should be in the form y = m x + b

Considering option A :

P = 25t + 100

Initial population = 100, population after 1 year = 125, after 2 years = 150

So, option A is correct.

Problem 3 :

The population of trees in a forest has been decreasing by 6 percent every 4 years. The population at the beginning of 2015 was estimated to be 14000. If P represents the population of trees t years after 2015, which of the following equations gives the population of trees over time ?

a)  P = 14000(0.06)^t/4       b)  14000 + 0.094 (4t) 

      c)  P = 100(0.94)^4t      d)  P = 14000(0.94)^t/4 

Solution :

Let initial population be 100%.

Every year it decreases 6%. Then 94% of population will be there after 4 years.

So, option D is correct. P = 14000(0.94)^t/4 

Problem 4 :

Jamie owes Tina some money and decides to pay her back in the following way. Tina receives 3 dollars the first day, 6 dollars the second day, 18 dollars the third day, and 54 dollars the fourth day. Which of the following best describes the relationship between time and the total amount of money (Cumulative) Tina has received from Jamie over the course of these four days ?

a)  Increasing linear         b) Decreasing linear 

c)  Exponential growth      d)  Exponential decay

Solution :

Amount she receives on the first day = $3

on the second day = $6

on the third day = 18

on fourth day = 54

the difference is not linear, it may be exponential function. Since it is increasing, it must be an exponential growth function.

Problem 5 :

Albert has large book collection. He decides to trade in two of his used books for one new book each month at a local bookstore. Which of the following best describes the relationship between time (in months) and the total number of books in Albert's collection ?

a)  Increasing linear         b) Decreasing linear 

c)  Exponential growth      d)  Exponential decay

Solution :

Every month number of books to be sold out = 2, number of new books can be purchased = 1

So, there will be 1 book will be more in every month. It is decreasing linear.

Problem 6 :

A scientist counts 80 cells in a petri dish and finds that each one splits into two new cells every hour. He uses the function A(t) = cr^t to calculate the total number of cells in the petri dish after t hours. Which of the following assigns the correct values to c and r ?

a) c = 40, r = 2     b) c = 80, r = 0.5

c)  c = 80, r = 1.5     d)  c = 80, r = 2

Solution :

A(t) = cr^t

A(t) = 80, common ratio is 2 for every one hour.

80 = c(2)^t

When t = 2

80 = c(4)

c = 40

option A is correct.

Problem 7 :

Of the following, which one would result in linear growth of the square footage of a store ?

a)  The owner increase the square footage by 0.75% each year.

b)  The owner increase the square footage by 5% each year.

c) The owner expands the store by 5% of the original square footage each year.

d)  The owner alternates between adding 200 square feet one year and 300 square feet the next year.

Solution :

5% of the original square foot, so option c is correct.

Problem 8 :

The value of a stock is going up by 200% every hour. Which of the following best describes the relationship between time (in hours) and the value of the stock ?

a)  Increasing linear         b) Decreasing linear 

c)  Exponential growth      d)  Exponential decay

Solution :

Every year 200% increases. So, it is exponential growth.