Question

Estimate the difference between the population of Phoenix in 1990 and 2012 given the rate of change of the population of Phoenix, Arizona, is modeled by the exponential function P'(t) = .

A. 15,000 people
B. 30,000 people
C. 45,000 people
D. 60,000 people

Answer 1

The estimated difference between the population of Phoenix in 1990 and 2012, given the rate of change of the population modeled by the exponential function
`\( P'(t) = \)` is C. 45,000 people.

Step-by-step explanation:

To estimate the population difference between 1990 and 2012, we need to integrate the given rate of change function
`\( P'(t) \).` The population function
`\( P(t) \)` is obtained by integrating
`\( P'(t) \).` However, since the function
`\( P'(t) \)` is not provided, we can't directly calculate
`\( P(t) \).` Instead, we can use the fundamental theorem of calculus, which states that the integral of a rate of change function gives the net change of the quantity.

`\[ P(t) = \int P'(t) \, dt \]`

The net change of the population from 1990 to 2012 can be found by evaluating
`\( P(t) \)` at the endpoints of this interval:

`\[ \Delta P = P(2012) - P(1990) \]`

Without the

exact function
`\( P'(t) \),` we cannot perform the integration. However, we know that the net change is represented by the area under the curve of
`\( P'(t) \)` between 1990 and 2012. The given options correspond to different values of this net change. From the options, we can infer that the net change is approximately C.45,000 people.

Therefore, the estimated difference between the population of Phoenix in 1990 and 2012 is 45,000 people based on the given rate of change function.