A thin, straight, uniform rod of length $\ell=1.00 \mathrm{~m}$ and mass $m=185 \mathrm{~g}$ hangs from a pivot at one end. What is its period for small-amplitude oscillations?

A gear is to be manufactured from iron powders. It is desired that it have a final density $90 \%$ that of cast iron, and it is known that the shrinkage in sintering will be approximately $5 \%$. For a gear that is $2.5$ in. in diameter and has a $0.75$-in. hub, what is the required press force?

An ornamental light bulb is designed by revolving the graph of y = 1/3x^½ - x^3/2, 0 ≤ x ≤ 1/3 about the x-axis, where x and y are measured in feet. Find the surface area of the bulb and use the result to approximate the amount of glass needed to make the bulb. (Assume that the glass is 0.015 inch thick.)

You purchase a set of 100 cube-shaped miniature magnets, each with an edge length of

$$\frac { 1 } { 8 }$$

inch. You arrange the cubes to form larger cubes, each one with a different edge length. How many cubes can you form? What is their total volume?

For a new punch press, company A charges $250,000 to deliver and install it. Company A has estimated that the machine will have operating and maintenance (O&M) costs of$4000 a year. You estimate an annual benefit of $89,000. Company B charges$205,000 to deliver and install the device. Company B has estimated O&M of the press at $4300 a year. You estimate an annual benefit of$86,000. Both machines will last 5 years and can be sold for $15,000. Use an interest rate of 12%. Which machine should your company buy?

For random variables X and Y, show that

$$\operatorname{Cov}(X+Y, X-Y)=\operatorname{Var}(X)-\operatorname{Var}(Y) .$$

Selby walked for a while at $4 \mathrm{mph}$ and then jogged the rest of the way at $8 \mathrm{mph}$. If she covered 56 miles in 10 hours, how far did she walk and how far did she jog?

Explain in your own words what the term random sample means.

Perform the following conversion : $7521 \mathrm{~m}$ to miles

The diffusion equation is valid when many elementary steps are taken in the time interval of interest, but the random walk calculation makes it possible to discuss distributions for short times as well as for long. Use the expression $P(n d)= N ! /\left(N-N_{B}\right) ! N_{B} ! 2^{N}$ to calculate the probability of being six paces from the origin (i.e. at x = 6d) after (a) four, (b) six, (c) twelve steps.

Find an equation of the curve that satisfies the given conditions. At each point (x, y) on the curve the slope is $2 x+1$; the curve passes through the point $(-3,0)$.

Congress proposes a law that states that the Speaker of the House of Representatives will also serve as the president's White House press secretary. The president is outraged by this proposal. What does he/she have the power to do?

The small circular piston of a hydraulic press has an area of $8.00 \mathrm{~cm}^2$. What must the area of the large piston be to exert a pressing force of $3650 \mathrm{~N}$ , if the applied force is $25.0 \mathrm{~N}$.

Find a formula for $\int_a^b c d x$, where $a, b$, and $c$ are positive and $a<b$.