Water is flowing into a cylindrical tank that is initially empty for $$30$$ minutes. The cylindrical tank has radius $$4$$ feet. The height of the water in the tank is changing at the rate

$$\frac{dh}{dt}=3\sqrt{t}\cos\left(0.04t\right)$$ feet per minute, where $$t$$ is measured in minutes.

$$\left(a\right)$$ Write an expression involving an integral that gives the height of the water in the tank as a function of time $$t.$$

$$\left(b\right)$$ At time $$t=12,$$ what is the height of the water in the tank?

$$\left(c\right)$$ At what rate is the volume of water in the tank changing at time $$t=12?$$

$$\left(d\right)$$ Find the average rate of change of the height of the water and the average rate of change of the volume of water in the tank over the interval $$0\le t\le12.$$

$$\left(e\right)$$ At the same time, water is entering a second cylindrical tank identical to the one above. The height of the water in this second tank is changing at the rate

$$R_2\left(t\right)=0.0015t^3$$ feet per minute. Let

$$R_1\left(t\right)=3\sqrt{t}\cos\left(0.04t\right)$$ and define

$$F\left(t\right)=R_1\left(t\right)-R_2\left(t\right).$$

Is there a time $$t,$$ $$6$$<$$t$$<$$25,$$ such that the heights of water in the two tanks are changing at the same rate? Use the Intermediate Value Theorem to justify your answer.

Section II Part A