Question

The world population was 2560 million people in 1950 and 3040 million in 1960 and can be modeled by the function p(t) 2560e0.017185t , where t is time in years after 1950 and p(t) is the population in millions. Determine the average rate of change of p(t) in millions of people per year, from 4 ≤ t ≤ 8. Round your answer to the nearest hundredth.

Answer 1

Answer:

Step-by-step explanation:

The average rate of change is the slope.  However, since this is not a linear function but an exponential one, we cannot determine the slope of the function, only the slope of the secant line that connects the x-values of 4 and 8.  In order to do that we need to find the corresponding y-coordinate that goes with each of those x-values and plug them into the slope function.  What we will get is a very loose interpretation of the rate of change of population, but that's all we have short of using calculus.  It would be a closer estimation if we looked at x values of 4 and 5, or eve 4 ad 4.5.  But that's not what we're being asked.  So let's get to it.

Sub first 4 in for x and then 8, to get each y value:

$P(4)=2560e^{(.017185)(4)}$ to get that P(4) = 2742.16

$P(8)=2560e^{(.017185)(8)}$ to get that P(8) = 2937.29

The coordinates for these are (4, 2742.16) and (8, 2937.29)

Plug into the slope formula:

$m=\frac{2937.29-2742.16}{8-4}$ which gives you, in decimal form rounded to the nearest hundredth,

48.78 million.

The interpretation of this value in our situation is that between 1954 and 1958 the population increased 48.78 million people per year.