In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.


a. What is the SAV ratio of a cube with side lengths a?

Calculating work for different springs Calculate the work required to stretch the following springs 0.5m from their equilibrium positions. Assume Hooke’s law is obeyed.

a. A spring that requires a force of 50 N to be stretched 0.2 m from its equilibrium position

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

a. The displacement between t=0 and t=5

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.

a. Determine the position function, for t≥0, using the antiderivative method

v(t) = 9−t² on [0, 4]; s(0)=−2

Critical depth A large tank has a plastic window on one wall that is designed to withstand a force of 90,000 N. The square window is 2 m on a side, and its lower edge is 1 m from the bottom of the tank.

a. If the tank is filled to a depth of 4 m, will the window withstand the resulting force?

"Determine whether the following statements are true and give an explanation or counterexample.

a. A pyramid is a solid of revolution. "

Find the area of the region (see figure) in two ways.

a. Using integration with respect to x.