Math 3112
Test 2
Fall 1996

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The test has 6 problems on 4 pages.
Read the problems carefully!

Problem 1 (15 points)   Oil is flowing through a cylindrical pipeline of radius 1 foot. Since oil is viscous and sticks to the pipe, the rate of oil flowing through the pipe varies with the distance from the center. Suppose the speed of the oil at a distance r feet from the center of the pipeline is given by (1-r³) feet per second. What is the rate (in cubic feet per second), at which the oil is flowing through the pipeline?

Problem 2 (15 points)   Compute $$\int_1^\infty \frac{dx}{x^2+1}$$

Problem 3 (20 points)   Are the following integrals converging or diverging? Explain your reasoning!

1.

$$\int_2^{\infty} \frac{2 dx}{x-1}$$

2.

$$\int_5^{\infty}\frac{dx}{x (\ln x)^2}$$

Problem 4 (15 points)   Find the volume of the object you obtain, when you rotate the ellipse $$\frac{x^2}{4} + \frac{y^2}{9}=1$$ about the x-axis.

Problem 5 (15 points)   A completely filled reservoir has a dam at one end. Find the total force of the water acting on the trapezoidally shaped dam. Recall that water weighs 62.4 lbs/ft³; force=pressure $\times$ area; pressure=weight per unit volume $\times$ depth.

Problem 6 (20 points)   Consider a group of people who have received treatment for a life-threatening disease. Let t be the survival time, the number of years a person lives after receiving treatment. The probability density function giving the distribution of t is given by

$$p(t)=\left\{ \begin{array}{cl} 0&\mbox{, if }t<0\\ C e^{-Ct}&\mbox{, if }t \geq 0 \end{array}\right.$$

for some positive constant C.

1.

Find the cumulative distribution function P(t) for the given distribution p(t).

2.

What is the practical meaning of the value of P(t) in this context?

3.

Suppose a patient has a 70% probability of surviving at least 3 years. Find the constant C.