Exercise 26

Consider the box with a sloped top that is given by the following description: the base is the rectangle $R = [1,4] \times [2,5]$, while the top is given by the plane $z = p(x,y) = 30 - x - 2y$.

1. Write an iterated integral whose value represents the volume under $p$ over the rectangle $R$.

2. Evaluate the iterated integral you determined in (a).

3. What is the exact average value of $p$ over $R$?

4. If you wanted to build a rectangular box (with an identical base) that has the same volume as the box with the sloped top described here, how tall would the rectangular box have to be?