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Chapter 3.1

Q. 3.1.6

Gray Wolf Population

Insatiable killer. That’s the reputation the gray wolf acquired in the United States in the nineteenth and early twentieth centuries. Although the label was undeserved, an estimated two million wolves were shot, trapped, or poisoned. By 1960, the population was reduced to 800 wolves. Figure 3.6 shows the rebounding population in two recovery areas after the gray wolf was declared an endangered species and received federal protection.

The exponential function

f(x)=1145 e^{0.0325 x}

models the gray wolf population of the Western Great Lakes, f (x), x years after 1978.

a. According to the model, what was the gray wolf population, rounded to the nearest whole number, of the Western Great Lakes in 2014?

b. Does the model underestimate or overestimate the gray wolf population of the Western Great Lakes in 2014? By how much?

3.6

Step-by-Step

Verified Solution

a. Because 2014 is 36 years after 1978 (2014 1978 = 36), we substitute 36 for x in the given function.

f(x)=1145 e^{0.0325 x}                This is the given function.

f(36)=1145 e^{0.0325(36)}                Substitute 36 fox x.

Perform this computation on your calculator.

Scientific calculator:      1145 \fbox{×} \fbox{(}.0325 \fbox{×} 36 \fbox{)} e^x \fbox{=}

Graphing calculator:
The display should be approximately 3689.181571. Thus,

f(36)=1145 e^{0.0325(36)} \approx 3689.

According to the model, the gray wolf population of the Western Great Lakes in 2014 was approximately 3689 wolves.

b. Figure 3.6 shows that the gray wolf population of the Western Great Lakes in 2014 was 3686 wolves. The model in part (a) indicates a population of 3689 wolves. Thus, the model overestimates the population by 3689 – 3686, or by 3 wolves.