What do you do to the first inequality to get the second inequality?

$$x + 8 \leq 11 ; x \leq 3$$

Tell whether each statement is sometimes, always, or never true. Pointillism is a technique in which tiny dots of complementary colors are combined to form a picture. Which postulate ensures that a line connecting two of these points also lies in the plane containing the points?

A trapezoidal channel has n = 0.022 and

$$S_0 = 0.0003$$

and is made in the shape of a half-hexagon for maximum efficiency. What should the length of the side of the hexagon be if the channel is to carry

$$225 ft^3/s$$

of water? What is the discharge of a semicircular channel of the same cross-sectional area and the same

$$S_0$$

and n?

Use slope to determine whether the points A(3, 5), B(-2,6), and C(-5,7) are on the same line. Justify your conclusion algebraically.

A diesel engine does not use spark plugs to ignite the fuel and air in the cylinders. Instead, the temperature required to ignite the fuel occurs because the pistons compress the air in the cylinders . Suppose air at an initial temperature of

$$21 ^ { \circ } \mathrm { C }$$

is compressed adiabatically to a temperature of

$$688 ^ { \circ } \mathrm { C }$$

Assume the air to be an ideal gas for which

$$\gamma = \frac { 7 } { 5 }$$

Find the compression ratio, which is the ratio of the initial volume to the final volume.

We want to position a space probe along a line that extends directly toward the Sun in order to monitor solar flares. How far from Earth’s center is the point on the line where the Sun’s gravitational pull on the probe balances Earth’s pull?

Two paths

$$\mathbf{r}_1(t)$$

and

$$\mathbf{r}_2(t)$$

intersect if there is a point P lying on both curves. We say that

$$\mathbf{r}_1(t)$$

and

$$\mathbf{r}_2(t)$$

collide if

$$\mathbf{r}_1(t_0) = \mathbf{r}_2(t_0)$$

at some time

$$t_0.$$

Which of the following statements are true? (a) If

$$\mathbf{r}_1(t)$$

and

$$\mathbf{r}_2(t)$$

intersect, then they collide. (b) If

$$\mathbf{r}_1(t)$$

and

$$\mathbf{r}_2(t)$$

collide, then they intersect. (c) Intersection depends only on the underlying curves traced by

$$\mathbf{r}_1$$

and

$$\mathbf{r}_2,$$

but collision depends on the actual parametrizations.

Two thousand raffle tickets are sold for $3 each. Three prizes will be awarded: one for$1000 and two for $500. Assume that the probability that any given ticket is selected for the$1000 prize is 1/2000 and the probability that any given ticket is selected for the $500 prize is 2/2000. Jeremy purchases one of these tickets. (a) Determine his expected value. (b) Determine the fair price of a ticket.

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. If

$$\mathbf v$$

is a vector in a vector space V and r and s are scalars such that

$$r \mathbf { v } = s \mathbf { v },$$

then r = s.

$$\text { Exhibit four nonisomorphic groups of order } 18$$

For the limit

$$lim x→∞ 2x+1/√x+1 = ∞$$

by finding a value of N that corresponds to M = 100.

A cylinder with rotational inertia

$$I_1 = 2.0 kg·m^2$$

rotates clockwise about a vertical axis through its center with angular speed

$$ω_1 = 5.0 rad/s.$$

A second cylinder with rotational inertia

$$I_2 = 1.0 kg·m^2$$

rotates counterclockwise about the same axis with angular speed

$$ω_2 = 8.0 rad/s.$$

If the cylinders couple so they have the same rotational axis what is the angular speed of the combination? What percentage of the original kinetic energy is lost to friction?

For the following system

$$\begin{gather*} A = \left(\begin{array}{cc}3 & -1 \\ 6 & \,\,\,\, 2\end{array}\right), \end{gather*}$$

use elementary row and column operations to simplify the task of evaluating the determinant by cofactor expansions. If ${A}$ is nonsingular, find ${A^{-1}}$.

A string with fixed ends is acted on by a periodic force $F ( x , t ) = F ( x ) \sin \omega t$ per unit mass, so

$$y _ { t t } = a ^ { 2 } y _ { x x } + F ( x ) \sin \omega t.$$

Substitute

$$y ( x , t ) = \sum _ { n = 1 } ^ { \infty } c _ { n } \sin \frac { n \pi x } { L } \sin \omega t$$

and

$$F ( x ) = \sum _ { n = 1 } ^ { \infty } F _ { n } \sin \frac { n \pi x } { L }$$

to derive the steady periodic solution

$$y ( x , t ) = \sum _ { n = 1 } ^ { \infty } \frac { F _ { n } \sin ( n \pi x / a ) \sin \omega t } { \omega _ { n } ^ { 2 } - \omega ^ { 2 } },$$

where $\omega _ { n } = n \pi a / L$. Hence resonance does not result if $\omega = \omega _ { n }$ but $F _ { n } = 0$.