2.7 Section Exercises

Verbal

1. With what kind of exponential model would half-life be associated? What role does half-life play in these models?

Show Solution

2. What is carbon dating? Why does it work? Give an example in which carbon dating would be useful.

3. With what kind of exponential model would doubling time be associated? What role does doubling time play in these models?

Show Solution

4. Define Newton’s Law of Cooling. Then name at least three real-world situations where Newton’s Law of Cooling would be applied.

5. What is an order of magnitude? Why are orders of magnitude useful? Give an example to explain.

Show Solution

An order of magnitude is the nearest power of ten by which a quantity exponentially grows. It is also an approximate position on a logarithmic scale; Sample response: Orders of magnitude are useful when making comparisons between numbers that differ by a great amount. For example, the mass of Saturn is 95 times greater than the mass of Earth. This is the same as saying that the mass of Saturn is about $\text{ }{10}^{\text{2}}\text{ }$ times, or 2 orders of magnitude greater, than the mass of Earth.

Numeric

6. The temperature of an object in degrees Fahrenheit after t minutes is represented by the equation $\text{ }T\left(t\right)=68{e}^{-0.0174t}+72.\text{ }$ To the nearest degree, what is the temperature of the object after one and a half hours?

For the following exercises, use the logistic growth model $\text{ }f\left(x\right)=\frac{150}{1+8{e}^{-2x}}.$

7. Find and interpret $\text{ }f\left(0\right).\text{ }$ Round to the nearest tenth.

Show Solution

$f\left(0\right)\approx 16.7;\text{ }$ The amount initially present is about 16.7 units.

8. Find and interpret $\text{ }f\left(4\right).\text{ }$ Round to the nearest tenth.

9. Find the carrying capacity.

Show Solution

10. Graph the model.

11. Determine whether the data from the table could best be represented as a function that is linear, exponential, or logarithmic. Then write a formula for a model that represents the data.

$x$	$f\left(x\right)$
–2	0.694
–1	0.833
0	1
1	1.2
2	1.44
3	1.728
4	2.074
5	2.488

Show Solution

exponential; $\text{ }f\left(x\right)={1.2}^{x}$

12. Rewrite $\text{ }f\left(x\right)=1.68{\left(0.65\right)}^{x}\text{ }$ as an exponential equation with base $\text{ }e\text{ }$ to five significant digits.

Technology

For the following exercises, enter the data from each table into a graphing calculator and graph the resulting scatter plots. Determine whether the data from the table could represent a function that is linear, exponential, or logarithmic.

13.

$x$	$f\left(x\right)$
1	2
2	4.079
3	5.296
4	6.159
5	6.828
6	7.375
7	7.838
8	8.238
9	8.592
10	8.908

Show Solution

14.

$x$	$f\left(x\right)$
1	2.4
2	2.88
3	3.456
4	4.147
5	4.977
6	5.972
7	7.166
8	8.6
9	10.32
10	12.383

15.

$x$	$f\left(x\right)$
4	9.429
5	9.972
6	10.415
7	10.79
8	11.115
9	11.401
10	11.657
11	11.889
12	12.101
13	12.295

Show Solution

16.

$x$	$f\left(x\right)$
1.25	5.75
2.25	8.75
3.56	12.68
4.2	14.6
5.65	18.95
6.75	22.25
7.25	23.75
8.6	27.8
9.25	29.75
10.5	33.5

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in $\text{ }t\text{ }$ years is modeled by the equation $\text{ }P\left(t\right)=\frac{1000}{1+9{e}^{-0.6t}}.$

17. Graph the function.

Show Solution

18. What is the initial population of fish?

19. To the nearest tenth, what is the doubling time for the fish population?

Show Solution

about $\text{ }1.4\text{ }$ years

20. To the nearest whole number, what will the fish population be after $\text{ }2\text{ }$ years?

21. To the nearest tenth, how long will it take for the population to reach $\text{ }900?$

Show Solution

about $\text{ }7.3\text{ }$ years

22. What is the carrying capacity for the fish population? Justify your answer using the graph of $\text{ }P.$

Extensions

23. A substance has a half-life of 2.045 minutes. If the initial amount of the substance was 132.8 grams, how many half-lives will have passed before the substance decays to 8.3 grams? What is the total time of decay?

Show Solution

$4\text{ }$ half-lives; $\text{ }8.18\text{ }$ minutes

24. The formula for an increasing population is given by $\text{ }P\left(t\right)={P}_{0}{e}^{rt}\text{ }$ where $\text{ }{P}_{0}\text{ }$ is the initial population and $\text{ }r>0.\text{ }$ Derive a general formula for the time t it takes for the population to increase by a factor of M.

25. Recall the formula for calculating the magnitude of an earthquake, $\text{ }M=\frac{2}{3}\mathrm{log}\left(\frac{S}{{S}_{0}}\right).$ Show each step for solving this equation algebraically for the seismic moment $\text{ }S.$

Show Solution

$\begin{array}{l}\text{ }M=\frac{2}{3}\mathrm{log}\left(\frac{S}{{S}_{0}}\right)\hfill \\ \mathrm{log}\left(\frac{S}{{S}_{0}}\right)=\frac{3}{2}M\hfill \\ \text{ }\frac{S}{{S}_{0}}={10}^{\frac{3M}{2}}\hfill \\ \text{ }S={S}_{0}{10}^{\frac{3M}{2}}\hfill \end{array}$

26. What is the y-intercept of the logistic growth model $\text{ }y=\frac{c}{1+a{e}^{-rx}}?\text{ }$ Show the steps for calculation. What does this point tell us about the population?

27. Prove that $\text{ }{b}^{x}={e}^{x\mathrm{ln}\left(b\right)}\text{ }$ for positive $\text{ }b\ne 1.$

Show Solution

Let $\text{ }y={b}^{x}\text{ }$ for some non-negative real number $\text{ }b\text{ }$ such that $\text{ }b\ne 1.\text{ }$ Then,

$\begin{array}{l}\mathrm{ln}\left(y\right)=\mathrm{ln}\left({b}^{x}\right)\hfill \\ \mathrm{ln}\left(y\right)=x\mathrm{ln}\left(b\right)\hfill \\ {e}^{\mathrm{ln}\left(y\right)}={e}^{x\mathrm{ln}\left(b\right)}\hfill \\ y={e}^{x\mathrm{ln}\left(b\right)}\hfill \end{array}$

Real-World Applications

For the following exercises, use this scenario: A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour.

28. To the nearest hour, what is the half-life of the drug?

29. Write an exponential model representing the amount of the drug remaining in the patient’s system after $\text{ }t\text{ }$ hours. Then use the formula to find the amount of the drug that would remain in the patient’s system after 3 hours. Round to the nearest milligram.

Show Solution

$A=125{e}^{\left(-0.3567t\right)};A\approx 43\text{ }$ mg

30. Using the model found in the previous exercise, find $\text{ }f\left(10\right)\text{ }$ and interpret the result. Round to the nearest hundredth.

For the following exercises, use this scenario: A tumor is injected with $\text{ }0.5\text{ }$ grams of Iodine-125, which has a decay rate of $\text{ }1.15%\text{ }$ per day.

31. To the nearest day, how long will it take for half of the Iodine-125 to decay?

Show Solution

about $\text{ }60\text{ }$ days

32. Write an exponential model representing the amount of Iodine-125 remaining in the tumor after $\text{ }t\text{ }$ days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days. Round to the nearest tenth of a gram.

33. A scientist begins with $\text{ }\text{250}\text{ }$ grams of a radioactive substance. After $\text{ }\text{250}\text{ }$ minutes, the sample has decayed to $\text{ }\text{32}\text{ }$ grams. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?

Show Solution

$f\left(t\right)=250{e}^{\left(-0.00914t\right)};\text{ }$ half-life: about $\text{ }\text{76}\text{ }$ minutes

34. The half-life of Radium-226 is $\text{ }1590\text{ }$ years. What is the annual decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

35. The half-life of Erbium-165 is $\text{ }\text{10}\text{.4}\text{ }$ hours. What is the hourly decay rate? Express the decimal result to four significant digits and the percentage to two significant digits.

Show Solution

$r\approx -0.0667,$ So the hourly decay rate is about $\text{ }6.67%$

36. A wooden artifact from an archeological dig contains 60 percent of the carbon-14 that is present in living trees. To the nearest year, about how many years old is the artifact? (The half-life of carbon-14 is $\text{ }\text{573}0\text{ }$ years.)

37. A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was $\text{ }1350\text{ }$ bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after $\text{ 3 }$ hours?

Show Solution

$f\left(t\right)=1350{e}^{\left(0.03466t\right)};\text{ }$ after 3 hours: $\text{ }P\left(180\right)\approx 691,200$

For the following exercises, use this scenario: A biologist recorded a count of $\text{ }360\text{ }$ bacteria present in a culture after $\text{ }5\text{ }$ minutes and $\text{ }1000\text{ }$ bacteria present after $\text{ }20\text{ }$ minutes.

38. To the nearest whole number, what was the initial population in the culture?

39. Rounding to six significant digits, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?

Show Solution

$f\left(t\right)=256{e}^{\left(0.068110t\right)};\text{ }$ doubling time: about $\text{ }10\text{ }$ minutes

For the following exercises, use this scenario: A pot of boiling soup with an internal temperature of $\text{ }\text{100°}\text{ }$ Fahrenheit was taken off the stove to cool in a $\text{ }\text{69° F}\text{ }$ room. After fifteen minutes, the internal temperature of the soup was $\text{ }\text{95° F}\text{.}$

40. Use Newton’s Law of Cooling to write a formula that models this situation.

41. To the nearest minute, how long will it take the soup to cool to $\text{ }\text{80° F?}$

Show Solution

about $\text{ }\text{88}\text{ }$ minutes

42. To the nearest degree, what will the temperature be after $\text{ }2\text{ }$ and a half hours?

For the following exercises, use this scenario: A turkey is taken out of the oven with an internal temperature of $\text{ }\text{165°F}\text{ }$ and is allowed to cool in a $\text{ }\text{75°F}\text{ }$ room. After half an hour, the internal temperature of the turkey is $\text{ }\text{145°F}\text{.}$

43. Write a formula that models this situation.

Show Solution

$T\left(t\right)=90{e}^{\left(-0.008377t\right)}+75,$ where $\text{ }t\text{ }$ is in minutes.

44. To the nearest degree, what will the temperature be after 50 minutes?

45. To the nearest minute, how long will it take the turkey to cool to $\text{ }\text{110° F?}$

Show Solution

about $\text{ }\text{113}\text{ }$ minutes

For the following exercises, find the value of the number shown on each logarithmic scale. Round all answers to the nearest thousandth.

46.

Number line to show log(x) is between -1 and 0.

47.

Number line to show log(x) is between 1 and 2.

Show Solution

$\mathrm{log}\left(x\right)=1.5;\text{ }x\approx 31.623$

48. Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper: $\text{ }{10}^{-10} \frac{W}{{m}^{2}},$ Vacuum: $\text{ }{10}^{-4}\frac{W}{{m}^{2}},$ Jet: $\text{ }{10}^{2} \frac{W}{{m}^{2}}$

49. Recall the formula for calculating the magnitude of an earthquake, $\text{ }M=\frac{2}{3}\mathrm{log}\left(\frac{S}{{S}_{0}}\right).\text{ }$ One earthquake has magnitude $\text{ }\text{3}.\text{9}\text{ }$ on the MMS scale. If a second earthquake has $\text{ }\text{75}0\text{ }$ times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.

Show Solution

MMS magnitude: $\text{ }5.82$

For the following exercises, use this scenario: The equation $\text{ }N\left(t\right)=\frac{500}{1+49{e}^{-0.7t}}\text{ }$ models the number of people in a town who have heard a rumor after t days.

50. How many people started the rumor?

51. To the nearest whole number, how many people will have heard the rumor after 3 days?

Show Solution

$N\left(3\right)\approx 71$

52. As $\text{ }t\text{ }$ increases without bound, what value does $\text{ }N\left(t\right)\text{ }$ approach? Interpret your answer.

For the following exercise, choose the correct answer choice.

53. A doctor and injects a patient with $\text{ }13\text{ }$ milligrams of radioactive dye that decays exponentially. After $\text{ }12\text{ }$ minutes, there are $\text{ }4.75\text{ }$ milligrams of dye remaining in the patient’s system. Which is an appropriate model for this situation?

$f\left(t\right)=13{\left(0.0805\right)}^{t}$
$f\left(t\right)=13{e}^{0.9195t}$
$f\left(t\right)=13{e}^{\left(-0.0839t\right)}$
$f\left(t\right)=\frac{4.75}{1+13{e}^{-0.83925t}}$

Show Solution

Exponential and Logarithmic Functions