Verify the moments of inertia for the solid of uniform density. Use a graphing utility to evaluate the triple integrals.

$$\begin{aligned} & I_x=\frac{1}{12} m\left(a^2+b^2\right) \\ & I_y=\frac{1}{12} m\left(b^2+c^2\right) \\ & I_z=\frac{1}{12} m\left(a^2+c^2\right)\end{aligned}$$

Find the center of mass of the solid of uniform density bounded by the graphs of the equations.
$x^2+y^2+z^2=25, z=4$ (the larger solid)

The gap between the earnings of men and women with equal education is narrowing but has not closed. Sample data for seven men and seven women with bachelor’s degrees are as follows. Data are shown in thousands of dollars.

$$\begin{matrix} \text{Men} & \text{Women}\\ \hline \text{35.6} & \text{49.5}\\ \text{80.5} & \text{40.4}\\ \text{50.2} & \text{32.9}\\ \text{67.2} & \text{45.5}\\ \text{43.2} & \text{30.8}\\ \text{54.9} & \text{52.5}\\ \text{60.3} & \text{29.8}\\ \end{matrix}$$

a. What is the median salary for men? For women? b. Use $\alpha =$ .05 and conduct the hypothesis test for identical population distributions. What is your conclusion?

Horizontal cross sections of a piece of ice that broke from a glacier are in the shape of a quarter of a circle with a radius of approximately 50 feet. The base is divided into 20 subregions, as shown in the figure. At the center of each subregion, the height of the ice is measured, yielding the following points in cylindrical coordinates.

$\left(5, \frac{\pi}{16}, 7\right)$,$\left(15, \frac{\pi}{16}, 8\right)$,$\left(25, \frac{\pi}{16}, 10\right)$,$\left(35, \frac{\pi}{16}, 12\right)$,$\left(45, \frac{\pi}{16}, 9\right)$, $\left(5, \frac{3 \pi}{16}, 9\right)$, $\left(15, \frac{3 \pi}{16}, 10\right)$, $\left(25, \frac{3 \pi}{16}, 14\right)$, $\left(35, \frac{3 \pi}{16}, 15\right)$, $\left(45, \frac{3 \pi}{16}, 10\right)$, $\left(5, \frac{5 \pi}{16}, 9\right)$, $\left(15, \frac{5 \pi}{16}, 11\right)$, $\left(25, \frac{5 \pi}{16}, 15\right)$, $\left(35, \frac{5 \pi}{16}, 18\right)$, $\left(45, \frac{5 \pi}{16}, 14\right)$, $\left(5, \frac{7 \pi}{16}, 5\right)$, $\left(15, \frac{7 \pi}{16}, 8\right)$, $\left(25, \frac{7 \pi}{16}, 11\right)$, $\left(35, \frac{7 \pi}{16}, 16\right)$, $\left(45, \frac{7 \pi}{16}, 12\right)$

Approximate the volume of the piece of ice.

Find the average value of f(x, y) over the plane region R. f(x, y) = x² + y² R: square with vertices (0, 0), (2, 0), (2, 2), (0, 2)

Sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration. ∫_0^6∫_0^6-x∫_0^6-x-y dz dy dx Rewrite using the order dy dx dz.

Use a computer algebra system or graphing utility to convert the coordinates of a point from one system to another among the rectangular, cylindrical, and spherical coordinate systems.

Horizontal cross sections of a piece of ice that broke from a glacier are in the shape of a quarter of a circle with a radius of approximately 50 feet. The base is divided into 20 subregions, as shown in the figure. At the center of each subregion, the height of the ice is measured, yielding the following points in cylindrical coordinates. $\left(5, \frac{\pi}{16}, 7\right),\left(15, \frac{\pi}{16}, 8\right),\left(25, \frac{\pi}{16}, 10\right),\left(35, \frac{\pi}{16}, 12\right),\left(45, \frac{\pi}{16}, 9\right)$, $\left(5, \frac{3 \pi}{16}, 9\right),\left(15, \frac{3 \pi}{16}, 10\right),\left(25, \frac{3 \pi}{16}, 14\right),\left(35, \frac{3 \pi}{16}, 15\right),\left(45, \frac{3 \pi}{16}, 10\right)$, $\left(5, \frac{5 \pi}{16}, 9\right),\left(15, \frac{5 \pi}{16}, 11\right),\left(25, \frac{5 \pi}{16}, 15\right),\left(35, \frac{5 \pi}{16}, 18\right),\left(45, \frac{5 \pi}{16}, 14\right)$, $\left(5, \frac{7 \pi}{16}, 5\right),\left(15, \frac{7 \pi}{16}, 8\right),\left(25, \frac{7 \pi}{16}, 11\right),\left(35, \frac{7 \pi}{16}, 16\right),\left(45, \frac{7 \pi}{16}, 12\right)$ (a) Approximate the volume of the piece of ice. (b) Ice weighs approximately 57 pounds per cubic foot. Approximate the weight of the piece of ice. (c) There are 7.48 gallons of water per cubic foot. Approximate the number of gallons of water in the piece of ice.

Use the graphical method to construct the shearforce and bending-moment diagrams for the beams shown in Figures . Label all significant points on each diagram, and identify the maximum moments (both positive ,negative) along with their respective locations. Clearly differentiate straight-line and curved portions of the diagrams.

An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About 40% break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay from another supplier She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks.

Suppose the new, expensive clay really is no better than her usual clay. What's the probability that this test convinces her to use it anyway? (Hint: Use a Binomial model.)

When sampling is done without replacement from a finite population, the hyper-geometric distribution is the exact probability distribution for the number of members sampled that have a specified attribute. The hypergeometric probability formula is

$$P(X=x)=\frac{\left(\begin{array}{c} N p \\ x \end{array}\right)\left(\begin{array}{c} N(1-p) \\ n-x \end{array}\right)}{\left(\begin{array}{l} N \\ n \end{array}\right)}$$

, where X denotes the number of members sampled that have the specified attribute, N is the population size, a is the sample size, and p is the population proportion. To illustrate, suppose that a customer purchases 4 fuses from a shipment of 250, of which 94% are not defective. Let a success correspond to a fuse that is not defective. a. Determine N, n, and p. b. Apply the hypergeometric probability formula to determine the probability distribution of the number of nondefective fuses that the customer gets. c. Obtain the binomial distribution with parameters n=4 and p=0.94. d. Compare the hypergeometric distribution that you obtained in part (b) with the binomial distribution that you obtained in part (c).

Researchers at Harris Interactive wondered if there was a difference between males and females in regard to some common annoyances. They asked a random sample of males and females the following question: "Are you annoyed by people who repeatedly check their mobile phones while having an in-person conversation?" Among the 540 males surveyed, 178 responded "Yes"; among the 560 females surveyed, 206 responded "Yes."

What is the response variable in the study?

A spaceship has length 120 m, diameter 25 m, and mass $4.0 \times 10^{3}$kg as measured by its crew. As the spaceship moves parallel to its cylindrical axis and passes us, we measure its length to be 90 m.

What do we measure the magnitude of its momentum to be?

A spaceship has length 120 m, diameter 25 m, and mass $4.0 \times 10^{3}$kg as measured by its crew. As the spaceship moves parallel to its cylindrical axis and passes us, we measure its length to be 90 m.

What do we measure its diameter to be?