Determine $\cos \theta$ where $\theta$ is the angle between u and v. $u = 2i - 3j + k, \ v = i - 2j + 3k$

Two strings, of tension T and mass densities $\mu_{1}$ and $\mu_{2}$, are connected together. Consider a traveling wave incident on the boundary. Find the ratio of the reflected amplitude to the incident amplitude, and the ratio of the transmitted amplitude to the incident amplitude, for the cases $\mu_{2} / \mu_{1}=0,0.25,1,4, \infty$.

The portable lighting equipment for a mine is located 100 meters from its dc supply source. The mine lights use a total of 5 kW and operate at 120 V dc. Determine the required cross-sectional area of the copper wires used to connect the source to the mine lights if we require that the power lost in the copper wires be less than or equal to 5 percent of the power required by the mine lights. Hint: Model both the lighting equipment and the wire as resistors.

What would be the maximum cost of a CFL such that the total cost (investment plus operating) would be the same for both CFL and incandescent 60-W bulbs? Assume the cost of the incandescent bulb is 25 cents and that electricity costs 10 cents/kWh. Calculate the cost for 1000 hours, as in the cost effectiveness of CFL example.

(a) The velocity of sound in a gas is proportional to the square root of the absolute temperature T. Use this fact, and the result of the previous problem, to show that when a thermal gradient exists in the vertical direction (z) sound waves will be turned initially with a radius of curvature $R=\frac{2 T}{d T / d z}$ (b) On a still day, the temperature of the atmosphere is found to decrease more or less linearly with height. Sketch the paths of "rays" of sound emitted from a source suspended high in the atmosphere. Assuming that the velocity of sound at ground level is 1110 ft/sec, estimate the horizontal distance at which an airplane flying at 15,000 ft firs becomes audible to an observer on the ground, if the temperature decreases by $1^{\circ} \mathrm{C}$ per 500-ft increase in altitude.

The area of a triangle is 96 square inches. Its height is 12 inches. Find the base.

$$A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right], \ B=\left[\begin{array}{ll} 1 & 2 \\ 2 & 4 \end{array}\right], \ C=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]$$

$$D=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{array}\right], \ E=\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 1 & 3 \end{array}\right]$$

,

$$F=\left[\begin{array}{lll} 1 & 2 & 1 \\ 0 & 3 & 2 \\ 0 & 0 & 1 \end{array}\right]$$

Determine whether the given matrix is singular or nonsingular. If a matrix M is singular, give all solutions of $M \mathbf{x}=\theta$. AB.

Let G = S$_{3}$, let H = A$_{3}$ and let V be the 3-dimensional $\mathbb{C}$H-module which affords the natural permutation representation of A$_{3}$. More explicitly, let V have basis $e_{1}$, $e_{2}$, $e_{3}$ and let $\sigma \in A_{3}$ act on V by

$$\sigma e_{i}=e_{\sigma(i)}.$$

Let 1 and (1 2) be coset representatives for the left cosets of A$_{3}$ in S$_{3}$ and write out the explicit matrices described in Theorem 11 for the action of S$_{3}$ on the induced module W, for each of the elements of S$_{3}$.

Calculate the character table of D$_{10}$. (This table contains nonreal entries.)

Let A = $\mathbb{Z}/4\mathbb{Z}$ and let G be the cyclic group of order 2 acting trivially on A. (a) Prove that $\left|C^{2}(G, A)\right|=2^{8}$. (b) Use the coboundary condition to show that $\left|B^{2}(G, A)\right|=2^{3}$. (c) Use the cocycle condition to show that $\left|Z^{2}(G, A)\right| \leq 2^{4}$. (d) Show that $\left|H^{2}(G, A)\right|=2$ by exhibiting two inequivalent extensions of G by A and their corresponding cocycles.

Solve the given system by forming $\mathbf{x}=A^{-1} \mathbf{b}$, where A is the coefficient matrix for the system.

$$\begin{aligned} x_{1}+x_{2} &=0 \\ 2 x_{1}+3 x_{2} &=4 \end{aligned}$$

Fusion probability is greatly enhanced when appropriate nuclei are brought close together, but mutual Coulomb repulsion must be overcome. This can be done using the kinetic energy of high-temperature gas ions or by accelerating the nuclei toward one another. (a) Calculate the potential energy of two singly charged nuclei separated by $1.00 \times 10^{-12} \mathrm{m}$ by finding the voltage of one at that distance and multiplying by the charge of the other. (b) At what temperature will atoms of a gas have an average kinetic energy equal to this needed electrical potential energy?

A skydiver will reach a terminal velocity when the air drag equals their weight. For a skydiver with high speed and a large body, turbulence is a factor. The drag force then is approximately proportional to the square of the velocity. Taking the drag force to be $F_{\mathrm{D}}=\frac{1}{2} \rho A v^{2}$ and setting this equal to the person's weight, find the terminal speed for a person falling "spread eagle." Find both a formula and a number for $v_{\mathrm{t}},$ with assumptions as to size.

In the following exercise, use a calculator to draw the graph of the piecewise-defined function and study the graph to evaluate the given limits.

$$g(x)=\left\{\begin{array}{ll} x^{3}-1, & x \leq 0 \\ 1, & x>0 \end{array}\right.$$

(a) $\lim _{x \rightarrow 0^{-}} g(x)$ (b) $\lim _{x \rightarrow 0^{+}} g(x)$