4.1E: Exponential Functions (Exercises)
Section 4.1 Exercise
For each table below, could the table represent a function that is linear, exponential, or neither?
1. 2.
3. 4.
5. 6.
7. A population numbers 11,000 organisms initially and grows by 8.5% each year. Write an exponential model for the population.
8. A population is currently 6,000 and has been increasing by 1.2% each day. Write an exponential model for the population.
9. The fox population in a certain region has an annual growth rate of 9 percent per year. It is estimated that the population in the year 2010 was 23,900. Estimate the fox population in the year 2018.
10. The amount of area covered by blackberry bushes in a park has been growing by 12% each year. It is estimated that the area covered in 2009 was 4,500 square feet. Estimate the area that will be covered in 2020.
11. A vehicle purchased for $32,500 depreciates at a constant rate of 5% each year. Determine the approximate value of the vehicle 12 years after purchase.
12. A business purchases $125,000 of office furniture which depreciates at a constant rate of 12% each year. Find the residual value of the furniture 6 years after purchase.
Find a formula for an exponential function passing through the two points.
13. \(\left(0,\; 6\right),\; (3,\; 750)\)
14. \(\left(0,\; 3\right),\; (2,\; 75)\)
15. \(\left(0,\; 2000\right),\; (2,\; 20)\)
16. \(\left(0,\; 9000\right),\; (3,\; 72)\)
17. \(\left(-1,\dfrac{3}{2} \right),\; \left(3,\; 24\right)\)
18. \(\left(-1,\dfrac{2}{5} \right),\; \left(1,10\right)\)
19. \(\left(-2,6\right),\; \left(3,1\right)\)
20. \(\left(-3,4\right),\; (3,\; 2)\)
21. \(\left(3,\; 1\right),\; (5,\; 4)\)
22. \(\left(2,5\right),\; (6,\; 9)\)
23. A radioactive substance decays exponentially. A scientist begins with 100 milligrams of a radioactive substance. After 35 hours, 50 mg of the substance remains. How many milligrams will remain after 54 hours?
24. A radioactive substance decays exponentially. A scientist begins with 110 milligrams of a radioactive substance. After 31 hours, 55 mg of the substance remains. How many milligrams will remain after 42 hours?
25. A house was valued at $110,000 in the year 1985. The value appreciated to $145,000 by the year 2005. What was the annual growth rate between 1985 and 2005? Assume that the house value continues to grow by the same percentage. What did the value equal in the year 2010?
26. An investment was valued at $11,000 in the year 1995. The value appreciated to $14,000 by the year 2008. What was the annual growth rate between 1995 and 2008? Assume that the value continues to grow by the same percentage. What did the value equal in the year 2012?
27. A car was valued at $38,000 in the year 2003. The value depreciated to $11,000 by the year 2009. Assume that the car value continues to drop by the same percentage. What was the value in the year 2013?
28. A car was valued at $24,000 in the year 2006. The value depreciated to $20,000 by the year 2009. Assume that the car value continues to drop by the same percentage. What was the value in the year 2014?
29. If $4,000 is invested in a bank account at an interest rate of 7 per cent per year, find the amount in the bank after 9 years if interest is compounded annually, quarterly, monthly, and continuously.
30. If $6,000 is invested in a bank account at an interest rate of 9 per cent per year, find the amount in the bank after 5 years if interest is compounded annually, quarterly, monthly, and continuously.
31. Find the annual percentage yield (APY) for a savings account with annual percentage rate of 3% compounded quarterly.
32. Find the annual percentage yield (APY) for a savings account with annual percentage rate of 5% compounded monthly.
33. A population of bacteria is growing according to the equation \(P(t)\; =\; 1600e^{0.21\; t}\), with t measured in years. Estimate when the population will exceed 7569.
34. A population of bacteria is growing according to the equation \(P(t)\; =\; 1200e^{0.17\; t}\), with t measured in years. Estimate when the population will exceed 3443.
35. In 1968, the U.S. minimum wage was $1.60 per hour. In 1976, the minimum wage was $2.30 per hour. Assume the minimum wage grows according to an exponential model \(w(t)\), where t represents the time in years after 1960. [UW]
a. Find a formula for \(w(t)\).
b. What does the model predict for the minimum wage in 1960?
c. If the minimum wage was $5.15 in 1996, is this above, below or equal to what the model predicts?
36. In 1989, research scientists published a model for predicting the cumulative number of AIDS cases (in thousands) reported in the United States: \(a\left(t\right)=155\left(\dfrac{t-1980}{10} \right)^{3}\), where \(t\) is the year. This paper was considered a “relief”, since there was a fear the correct model would be of exponential type. Pick two data points predicted by the research model \(a(t)\) to construct a new exponential model \(b(t)\) for the number of cumulative AIDS cases. Discuss how the two models differ and explain the use of the word “relief.” [UW]
37. You have a chess board as pictured, with squares numbered 1 through 64. You also have a huge change jar with an unlimited number of dimes. On the first square you place one dime. On the second square you stack 2 dimes. Then you continue, always doubling the number from the previous square. [UW]
a. How many dimes will you have stacked on the 10th square?
b. How many dimes will you have stacked on the nth square?
c. How many dimes will you have stacked on the 64th square?
d. Assuming a dime is 1 mm thick, how high will this last pile be?
e. The distance from the earth to the sun is approximately 150 million km. Relate the height of the last pile of dimes to this distance.
- Answer
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1. Linear
3. Exponential
5. Neither
7. \(P(t) = 11,000(1.085)^t\)
9. 47622 Fox
11. $17561.70
13. \(y =. 6(5)^x\)
15. \(y = 2000(0.1)^x\)
17. \(y = 3(2)^x\)
19. \(y = (\dfrac{1}{6})^{-\dfrac{3}{5}}(\dfrac{1}{6})^{\dfrac{x}{5}} = 2.93(0.699)^x\)
21. \(y = \dfrac{1}{8}(2)^x\)
23. 34.32 \(mg\)
25. 1.39%; $155,368.09
27. $4,813.55
29. Annual \(\approx\) $7353.84
Quarterly \(\approx\) $7,469.63
Monthly \(\approx\) $7,496.71
Continuously \(\approx\) $7,510.4431. 3.03%
33. 7.4 years
35a. \(w(t) = (1.113)(1.046)^t\)
b. $1.11
c. Below what the model predicts \(\approx\) $5.70