Marginal cost function graph, help!?

Question

http://math.msu.edu/%7eglashow1/E3_US07.pdf

For 7a

how did they get the (.5 $/toothbrush x 100 toothbrushes)

For 7b

How did they get ($50)(23 + 1/2(7))

Answer 1

Those questions appear to correspond to problem #4, not #7.

4a) (.5 $/toothbrush x 100 toothbrushes) is the area of each gridded box on the graph. The y-axis has grid lines every half-dollar, and the x-axis has grid lines every 100 toothbrushes.

Each grid box on the graph, therefore is $0.50 * 100.

4b) The $50 is the same as the value in the previous question -- the area of the gridded boxes on the graph. $0.50 * 100 = $50.

23 is the count of boxes between 0 and 500 toothbrushes that are entirely below the curve, and 7 is the count of boxes that are partially below the curve. (They're labeled "w" (whole) and "p" (partial) on the graph, respectively.)

The job is to estimate the integral from 0 to 500. The equation takes the number of "full" boxes, plus half the number of "partial" boxes (23 + (1/2)*7), and multiplies that by the area represented by one box ($0.50*100 = $50).

Obviously, since every "partial" box is not EXACTLY half-above and half-below the line, it's just an approximation. But it is a decent approximation of the area below the curve.