WORD PROBLEMS INVOLVING EXPONENTIAL EQUATIONS

Problem 1 :

The first year of a charity walk event had an attendance of 500. The attendance y increases by 5% each year.

A. Write an exponential growth function to represent this situation.

B. How many people will attend in the 10th year?

Solution:

A. 

y = A(1 + r)^x

Where A = first year attendance

r = 5% = 0.05

attendance after 't' years.

y = 500(1 + 5%)^t

y = 500(1 + 0.05)^t

y = 500(1.05)^t

B.

When x = 10

y = 500(1.05)¹⁰

y = 814 people

Problem 2 :

The population of a small town was 3600 in 2005. The population increases by 4% annually.

A. Write an exponential growth function to represent this situation.

B. What will the population be in 2025? Round your answer to the nearest person.

Solution:

A.

After 1 year, population will be

= 3600 + 3600 × 0.04

After 2 years, population will be

= 3600 + 3600 × 0.04 + (3600 + 3600 × 0.04) × 0.04

= 3600(1 + 0.04)²

After 't' years, population will be 3600(1 + 0.04)^t

So, y = 3600(1 + 0.04)^t

B. 

t in 2025

2025 - 2005 = 20 years

Population will be,

= 3600(1 + 0.04)²⁰

= 7888 people

Problem 3 :

Your starting salary at a new company $34,000 and it increases by 2.5% each year.

A. Write an exponential growth function to represent this situation.

B. What will you salary be in 5 years? Round your answer to the nearest dollar.

Solution:

Given, initial salary = $34,000

Increased rate = 2.5%

A.

y = 34000(1 + 2.5%)^t

y = 34000(1 + 0.025)^t

= 34000(1.025)^t  

B.

t = 5

y = 34000(1.025)⁵

y = $38470

Problem 4 :

In 2010 an item cost $9.00. The price increase by 1.5% each year.

A. Write an exponential growth function to represent this situation.

B. How much will it cost in 2030? Round your answer to the nearest cent.

Solution:

A.

y = 9.00(1 + 1.5%)^t

t = time in years

y = 9.00(1 + 0.015)^t

y = 9.00(1.015)^t

B.

At 2030, t = 20 years

y = 9.00(1.015)20

= $12.12

At 2030, it will cost $12.12.

Problem 5 :

The yearly profits of a company is $25,000. The profits have been decreasing by 6% per year.

A. Write an exponential decay function to represent this situation.

B. What will be the profits in 8 years? Round your answer to the nearest dollar.

Solution:

A.

y = 25000(1 - 6%)^t

y = 25000(1 - 0.06)^t

y = 25000(0.94)^t

B.

x = 8

y = 25000(0.94)⁸

y = $15239

Problem 6 :

You bought $2000 worth of stocks in 2012. The value of the stocks has been decreasing by 10% each year.

A. Write an exponential decay function to represent this situation.

B. What will your stock be worth in 2017? Round your answer to the nearest cent.

Solution:

A.

y = 2000(1 - 10%)^t

= 2000(1 - 0.1)^t

y = 2000(0.95)^t

B.

2017 - 2012 = 5

y = 2000(0.95)⁵

= $1180.98

So, the stack will be worth $1180.98 in 2017.

Problem 7 :

Your car cost $42,500 when you purchased it in 2015. The value of the car decreases by 15% annually.

A. Write an exponential decay function to represent this situation.

B. How much will your car be worth in 2022? Round your answer to the nearest dollar.

Solution:

A.

y = 42500(1 - 15%)^t

= 42500(1 - 0.15)^t

y = 42500(0.85)^t

B.

2022 - 2015 = 7 years

x = 7

y = 42500(0.85)⁷

y = $13624.5

Problem 8 :

A piece of land was purchased for $65,000. The value of the land has slowly been decreasing by 1% annually.

A. Write an exponential decay function to represent this situation.

B. How much will the land be worth in 20 years? Round your answer to the nearest dollar.

Solution:

A.

y = 65000(1 - 1%)^t

= 65000(1 - 0.01)^t

y = 65000(0.99)^t

B.

x = 20 years

y = 65000(0.99)²⁰

y = $53164