Section II Part A — Full Practice Exam 3

Questions in this quiz (2)

1. 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.

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2. The depth of water in tank $$A,$$ in inches, is modeled by a differentiable and increasing function $$h$$ for $$0\le t\le10,$$ where $$t$$ is measured in minutes. Values of $$h\left(t\right)$$ for selected values of $$t$$ are given in the table above.$$\left(a\right)$$ Use the data in the table to find an approximation for $$h'\left(6\right).$$ Show the computations that lead to your answer. Indicate units of measure.$$\left(b\right)$$ Approximate the value of $$\int_0^{10}h\left(t\right)\,dt$$ using a right Riemann sum with the four subintervals indicated by the data in the table. Is this approximation greater than or less than $$\int_0^{10}h\left(t\right)\,dt?$$$$\left(c\right)$$ The depth of water in tank $$B,$$ in inches, is modeled by the function $$g\left(t\right)=3.2+17.5\sqrt{\sin\left(0.16t\right)}$$ for $$0\le t\le10,$$ where $$t$$ is measured in minutes. Find the average depth of the water in tank $$B$$ over the interval $$0\le t\le10.$$ Is this value greater than or less than the average depth of the water in tank $$A$$ over the interval $$0\le t\le10?$$$$\left(d\right)$$ According to the model given in part $$\left(c\right),$$ is the depth of the water in tank $$B$$ increasing or decreasing at time $$t=6?$$ Give a reason for your answer.

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