CPM Homework Help : APCALC Problem 7-33

In order to plan for the future water supply, the average population of the world is needed. If the world population in 2015 was about $7.3$ billion people, determine the average population of the Earth during the next $30$ years. Assume the population, in billions, $t$ years from 2015 is projected to be:

$P(t)=7.3e^{0.014t}$

Hint:

You are asked to determine the average value of $P(t)$ , given the $P(t)$ function itself. Which variation of Mean Value will you use?

Math Notes:

Average (Mean) Values

To calculate the mean (average) value of a finite set of items, add up the values of items and divide by the number of items.

Integrals help us add over a continuous interval. Therefore, for any continuous function $f$ :

$\frac{\int_a^bf(x)dx}{b-a}=$ mean value of $f$ over $[a, b]$

First quadrant, bell curve labeled, f of x, left end point on the y axis, labeled, a, right end point labeled, b, dashed horizontal segment, about 1 fourth up from x axis to peak, labeled average, & shaded rectangle between, A & b, segment & x axis.

Since $\int _ { a } ^ { b } f ( x ) d x$ , we can also calculate the average value of any function $f$ using its antiderivative $F$ . Its average slope gives the average rate of change of $F$ , which is the same as the average value of $f.$

$\frac{\int_a^bf(x)dx}{b-a}=\frac { F ( b ) - F ( a ) } { b - a }=$ mean rate of change of $F ′$ over $[a, b]$

First quadrant, 2 tick marks on x axis, first at the origin labeled, A, second almost to the right end, labeled b, Increasing curve labeled, capital F of x, starting at the origin, changing from concave up to concave down, in center of quadrant, ending at point corresponding to, b, almost at the top, with dashed segment labeled, m = average, from origin to end point of curve.