sectioN exercises 551

For the following exercises, use this scenario: A biologist recorded a count of 360 bacteria present in a culture after 5 minutes and 1,000 bacteria present after 20 minutes.

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

40. 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?

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

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

42. To the nearest minute, how long will it take the soup to cool to 80° F?

43. To the nearest degree, what will the temperature be after 2 and a half hours?

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

44. Write a formula that models this situation. 45. To the nearest degree, what will the temperature be after 50 minutes?

46. To the nearest minute, how long will it take the turkey to cool to 110° F?

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

47.

0–1–2–3–4–5 1 2 3 4

log (x)

5

48.

0–1–2–3–4–5 1 2 3 4

log (x)

5

49. Plot each set of approximate values of intensity of sounds on a logarithmic scale: Whisper: 10−10 W ___ m2 ,

Vacuum: 10−4 W ___ m2 , Jet: 10 2 W ___ m2

50. Recall the formula for calculating the magnitude of an earthquake, M = 2 __ 3 log S __ S0 . One earthquake has magnitude 3.9 on the MMS scale. If a second earthquake has 750 times as much energy as the first, find the magnitude of the second quake. Round to the nearest hundredth.

For the following exercises, use this scenario: The equation N(t) = 500 _ 1 + 49e−0.7t

models the number of people in a town who have heard a rumor after t days.

51. How many people started the rumor? 52. To the nearest whole number, how many people will have heard the rumor after 3 days?

53. As t increases without bound, what value does N(t) approach? Interpret your answer.

For the following exercise, choose the correct answer choice.

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

a. f (t) = 13(0.0805)t b. f (t) = 13e0.9195t c. f (t) = 13e(−0.0839t) d. f (t) = 4.75 __________ 1 + 13e−0.83925t

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SECTION 6.8 sectioN exercises 561

6.8 SeCTIOn exeRCISeS

veRbAl

1. What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.

2. What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?

3. What is regression analysis? Describe the process of performing regression analysis on a graphing utility.

4. What might a scatterplot of data points look like if it were best described by a logarithmic model?

5. What does the y-intercept on the graph of a logistic equation correspond to for a population modeled by that equation?

gRAPhICAl

For the following exercises, match the given function of best fit with the appropriate scatterplot in Figure 7 through Figure 11. Answer using the letter beneath the matching graph.

1

2 4 6 8

10 12 14 16

y

x 2 3 4 5 6 7 8 9 10

(a)

Figure 7

1

2 4 6 8

10 12 14 16

y

x 2 3 4 5 6 7 8 9 10

(b)

Figure 8

2 4 6 8

10 12 14 16

y

1 2 3 4 5 6 7 8 9 10 x

(c)