Chapter 1.4, Problem 232E

A horizontal cylindrical tank has cross-sectional area $A\left(x\right)=4\left(6x-x^2\right)m^2$ at height x meters above the bottom when $x\le 3$ .

a. The volume V between heights a and b is ${\displaystyle {\int }_a^{b}A\left(x\right)dx}$ . Find the volume at heights between 2 m and 3 m.

b. Suppose that oil is being pumped into the tank at a rate of 50 L/min. Using the chain rule, $\frac{dx}{dt}=\frac{dx}{dV}\frac{dV}{dt}$ , at how many meters per minute is the height of oil in the tank changing, expressed in terms of x, when the height is at x meters?

c. How long does it take to fill the tank to 3 m starting from a fill level of 2 m?