Worksheet: Applications of Exponential Functions

In this worksheet, we will practice solving real-world problems involving exponential functions.

Q1:

A population of bacteria decreases as a result of a chemical treatment. The population 𝑑 hours after the treatment was applied can be modeled by the function 𝑃(𝑑), where 𝑃(𝑑)=6,000Γ—(0.4).

What was the population when the chemical was first applied?

What is the rate of population decrease?

  • A4% per hour
  • B1.4% per hour
  • C1.6% per hour
  • D6% per hour
  • E60% per hour

Q2:

A new antibiotic is being tested in a laboratory. A population of bacteria treated with the antibiotic decreases by one-third every hour. The initial population was 240 bacteria. Write an equation to find 𝑃, the number of bacteria left after 𝑑 hours.

  • A 𝑃 = 2 4 0 ο€Ό 1 3 𝑑 
  • B 𝑃 = 2 4 0 ο€Ό 2 3  
  • C 𝑃 = 2 4 0 ο€Ό 2 3 𝑑 
  • D 𝑃 = 2 4 0 ο€Ό 1 3  
  • E 𝑃 = 2 4 0 ( 3 ) 

Q3:

A start-up company noticed that the number of those who use its product doubles every month. This month, they had 4,000 users. Assuming this trend continues, write an equation that can be used to calculate π‘ˆ, the number of users in π‘š months’ time.

  • A π‘ˆ = 4 , 0 0 0 ( 3 ) 
  • B π‘ˆ = 4 , 0 0 0 ( π‘š ) 
  • C π‘ˆ = 4 , 0 0 0 ( π‘š ) 
  • D π‘ˆ = 4 , 0 0 0 ( 2 ) 
  • E π‘ˆ = 4 , 0 0 0 ( 2 π‘š )

Q4:

The number of people infected with a virus is increasing at a rate of 17% a year. In the last year, 12,500 people were infected with the virus.

Write an equation that can be used to calculate 𝑃, the number of people expected to be infected with the virus in the next π‘š months.

  • A 𝑃 = 1 2 , 5 0 0 ( 0 . 8 3 ) 
  • B 𝑃 = 1 2 , 5 0 0 ( 0 . 8 3 ) ο‘‚  
  • C 𝑃 = 1 2 , 5 0 0 ( 1 . 1 7 ) ο‘‚  
  • D 𝑃 = 1 2 , 5 0 0 ( 0 . 1 7 ) ο‘‚  
  • E 𝑃 = 1 2 , 5 0 0 ( 1 . 1 7 ) 

How many people would be expected to catch the virus in the next seven months? Give your answer to the nearest one hundred people.

  • A37,500
  • B13,700
  • C11,200
  • D13,600
  • E4,400

Q5:

The number of people visiting a museum is decreasing by 3% a year. This year, there were 50,000 visitors. Assuming the decline continues, write an equation that can be used to find 𝑉, the number of visitors there will be in 𝑑 years’ time.

  • A 𝑉 = 5 0 , 0 0 0 ( 0 . 9 7 ) 
  • B 𝑉 = 5 0 , 0 0 0 ( 1 . 0 3 ) 
  • C 𝑉 = 5 0 , 0 0 0 ( 0 . 0 3 ) 
  • D 𝑉 = 5 0 , 0 0 0 ( 0 . 7 ) 
  • E 𝑉 = 5 0 , 0 0 0 ( 3 ) 

Q6:

The number of marine organisms in a pool, 𝑦, after 𝑛 weeks is given by the formula 𝑦=4,344ο€Ό12. After how many weeks are there 1,086 marine organisms in the pool?

Q7:

A microorganism reproduces by binary fission, where every hour each cell divides into two cells. Given that there were 15,141 cells to begin with, determine how many cells there were after 5 hours.

Q8:

The population of a town doubles every 50 years. How long does it take for the population to triple? Round your answer to two decimal places.

Q9:

Which is a higher annual rate and by how much: 18.2% per year compounded weekly or 18.5% per year compounded quarterly?

  • A 1 8 . 5 % quarterly, better by about 0.1%
  • B 1 8 . 2 % weekly, better by about 10%
  • C 1 8 . 5 % quarterly, better by about 0.01%
  • D 1 8 . 5 % quarterly, better by about 10%
  • E 1 8 . 2 % weekly, better by about 0.1%

Q10:

A poorly performing fund is losing 10% of its value every week. Today, the value of investing in the fund is $5,200.

Write an equation that can be used to calculate 𝐼, the value, in dollars, of an investment 𝑑 days ago.

  • A 5 , 2 0 0 = 𝐼 ( 1 . 1 ) 
  • B 5 , 2 0 0 = 𝐼 ( 0 . 9 ) 
  • C 5 , 2 0 0 = 𝐼 ( 0 . 1 )  
  • D 5 , 2 0 0 = 𝐼 ( 1 . 1 )  
  • E 5 , 2 0 0 = 𝐼 ( 0 . 9 )  

What was the value of an investment 10 days ago? Give your answer to the nearest dollar.

Q11:

The value of a used car depreciates at a rate of 14% every year. If the car was bought for $15,000 in February 2017, how much would it be worth in February 2023? Give your answer to the nearest one hundred dollars.

Q12:

In January 2009, Elizabeth invested $2,500 in an account that pays an annual interest rate of 4%. If she made no further deposits or withdrawals, how much money was in the account in January 2017? Give your answer to the nearest cent if necessary.

Q13:

The value of an antique vase increases at a rate of 7% a year. If the vase was valued at $1,2005 years ago, what would its current value be? Give your answer to the nearest ten dollars if necessary.

Q14:

A radioactive element decays at a rate of 6% per hour. At midday, there were 45 g of the element in a sample.

Write an equation that can be used to calculate π‘š, the mass of the element left 𝑑 hours after midday.

  • A π‘š = 4 5 ( 0 . 9 4 ) 
  • B π‘š = 4 5 ( 0 . 6 ) 
  • C π‘š = 4 5 ( 𝑑 )  οŽ–  οŠͺ
  • D π‘š = 4 5 ( 1 . 0 6 ) 
  • E π‘š = 4 5 ( 0 . 0 6 ) 

How much of the element will be left at 4:45 pm? Give your answer in grams to one decimal place.

How much was there at 10 am? Give your answer in grams to one decimal place.

Q15:

The value, 𝑉(𝑑) dollars, of a property 𝑑 years from now can be modeled by the function 𝑉(𝑑)=300,000Γ—1.075.

What is the property’s value now?

What is the rate of increase in the property’s value?

  • A1.075% per year
  • B1.75% per year
  • C7.5% per year
  • D75% per year
  • E0.075% per year

Q16:

Different solutions were prepared by pouring different amounts of a blue ink in a beaker and adding water to obtain 250 mL of solution. Using a light source and a light detector, the light transmitted through the beaker was measured as a function of the concentration of the solution. The given figure shows the corresponding data for the transmittance as a percentage. A concentration of 0 corresponds to pure water, and the measured value of light transmitted through the beaker containing only water was used as reference to work out the transmittance as a percentage.

Which of the following expressions for the transmittance 𝑇 percentage as a function of the concentration 𝑐 in molars (M) does NOT correspond to the graph?

  • A 𝑇 = 1 0 0 β‹… 0 . 5   
  • B 𝑇 = 1 0 0 β‹… 2    l n l n    
  • C 𝑇 = 1 0 0 β‹… 𝑒   οŽ–  οŠͺ οŠͺ 
  • D 𝑇 = 1 0 0 β‹… 𝑒   οŽ–   
  • E 𝑇 = 1 0 0 β‹… 2    

Giving your answer accurate to two significant figures, calculate the solution concentration that gives a transmittance of 68%.

Q17:

The given table shows the values of the function 𝑓 at various inputs.

π‘š 0 2 4 6
𝑓 ( π‘š ) 230 140 85 52

Which expression for 𝑓(π‘š) best fits the data?

  • A 2 3 0 ( 1 . 1 2 )   
  • B 2 3 0 ( 0 . 7 9 ) 
  • C 2 3 0 ( 0 . 7 9 )  
  • D 2 3 0 ( 1 . 1 2 ) 
  • E 2 3 0 ( 0 . 7 9 )   

Q18:

The US Census is taken every ten years. The population of Texas was 3.05 million in 1900 and 20.9 million in 2000. By modeling the population growth as exponential, answer the following questions.

Write an exponential function in the form 𝑃(𝑑)=π‘ƒπ‘˜οŠ¦οŒ½ to model the population of Texas, in millions, 𝑑 decades after 1,900. Round your value of π‘˜ to 3 decimal places if necessary.

  • A 𝑃 ( 𝑑 ) = 3 . 7 
  • B 𝑃 ( 𝑑 ) = 1 . 2 1 2 ( 3 . 0 5 ) 
  • C 𝑃 ( 𝑑 ) = 1 . 2 1 2 ( 𝑑 )  οŽ–  
  • D 𝑃 ( 𝑑 ) = 3 . 0 5 ( 1 . 2 1 2 ) 
  • E 𝑃 ( 𝑑 ) = 3 . 0 5 ( 𝑑 )  οŽ–   

According to the model, what was the population of Texas in 1950? Give your answer to three significant figures.

Rewrite your function in the form 𝑃(𝑦)=𝑃(𝑏), where 𝑦 is the time in years after 1900. Round your value of 𝑏 to 4 decimal places.

  • A 𝑃 ( 𝑦 ) = 1 . 0 1 9 4 ( 3 . 0 5 ) 
  • B 𝑃 ( 𝑦 ) = 3 . 0 5 ( 1 . 0 1 9 4 )  
  • C 𝑃 ( 𝑦 ) = 3 . 0 5 ( 1 . 0 1 9 4 ) 
  • D 𝑃 ( 𝑦 ) = 0 . 0 1 9 4 ( 3 . 0 5 ) 
  • E 𝑃 ( 𝑦 ) = 3 . 0 5 ( 0 . 0 1 9 4 ) 

Q19:

We can use exponential functions to model situations in which a quantity builds up to a limiting value. Which graph demonstrates this build-up?

  • A
  • B
  • C
  • D

Q20:

Consider 𝑓(𝑑)=25β‹…2.

Using π‘Ž=𝑒()ln, write 𝑓(𝑑) in the form 𝑓(𝑑)=π΄π‘’οŒΌο, with 𝐴 and 𝑐 as constants.

  • A 𝑓 ( 𝑑 ) = 𝑒 (   )  l n
  • B 𝑓 ( 𝑑 ) = 2 5 𝑒 l n 
  • C 𝑓 ( 𝑑 ) = 𝑒 (  )  l n
  • D 𝑓 ( 𝑑 ) = 2 5 𝑒 (  )  l n
  • E 𝑓 ( 𝑑 ) = 2 5 𝑒 (  )  l n

Using the fact that if 𝑏>0, then lnlnο€Ή2=ο€Ήπ‘ο…οοŒΌο, write 𝑓(𝑑) in the form 𝑓(𝑑)=π΄π‘οŒΌο with 𝑏=10.

  • A 𝑓 ( 𝑑 ) = 2 5 β‹… 1 0 (   )  l n
  • B 𝑓 ( 𝑑 ) = 2 5 β‹… 1 0 ο€»   l n l n   
  • C 𝑓 ( 𝑑 ) = 2 5 β‹… 1 0 ο€»   l n l n   
  • D 𝑓 ( 𝑑 ) = 1 0 ο€»   l n l n   
  • E 𝑓 ( 𝑑 ) = 2 5 β‹… 1 0 (  )  l n

Q21:

A population of bacteria doubles in number every 5 minutes. If the population is one at 15:00, what would the population be at 16:00?

  • A1,024
  • B8,192
  • C2,048
  • D4,096
  • E512

Q22:

If $800 is earning interest semiannually at 2% per annum, what is the amount after 𝑛 years?

  • A 8 0 0 ( 0 . 0 2 )  
  • B 8 0 0 ( 1 . 0 2 ) 
  • C 8 0 0 ( 1 . 0 2 )  
  • D 8 0 0 ( 1 . 0 1 )  
  • E 8 0 0 ( 1 . 0 1 ) 

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