Use a graphing utility to find the magnitude and direction angles of the resultant of forces $\mathbf{F}_1$ and $\mathbf{F}_2$ with initial points at the origin. The magnitude and terminal point of each vector are given.

Vector	Magnitude	Terminal Point
$\mathbf{F}_1$	$50 \mathrm{lb}$	$(10,5,3)$
$\mathbf{F}_2$	$80 \mathrm{lb}$	$(12,7,-5)$

The magnitude of the force $\mathbf{F}$ is $0.2 \mathrm{~N}$ and its direction cosines are $\cos \theta_x=0.727, \cos \theta_y=-0.364$, and $\cos \theta_z=0.582$. Determine the magnitude of the moment of $\mathbf{F}$ about the axis $A B$ of the spool.

Table of yearly discharge (cubic kilometers of water) of the Mississippi river gives the volume of water discharged by the Mississippi River into th Gulf of Mexico for each year from 1954 to 2001. The units are cubic kilometers of water-the Mississippi is a big river.

(a) Make a graph of the distribution of water volume. Describe the overall shape of the distribution and any outliers.

(b) Based on the shape of the distribution, do you expect the mean to be close to the median, clearly less than the median, or clearly greater than the median? Why? Find the mean and the median to check your answer.

$$\begin{array}{cc|cc|cc} \text { YEAR } & \text { DISCHARGE } & \text { YEAR } & \text { DISCHARGE } & \text { YEAR } & \text { DISCHARGE } \\ \hline 1954 & 290 & 1970 & 540 & 1986 & 600 \\ \hline 1955 & 420 & 1971 & 480 & 1987 & 450 \\ \hline 1956 & 390 & 1972 & 600 & 1988 & 420 \\ \hline 1957 & 610 & 1973 & 880 & 1989 & 630 \\ \hline 1958 & 550 & 1974 & 710 & 1990 & 680 \\ \hline 1959 & 440 & 1975 & 670 & 1991 & 700 \\ \hline 1960 & 470 & 1976 & 420 & 1992 & 510 \\ \hline 1961 & 600 & 1977 & 430 & 1993 & 900 \\ \hline 1962 & 550 & 1978 & 560 & 1994 & 640 \\ \hline 1963 & 360 & 1979 & 800 & 1995 & 590 \\ \hline 1964 & 390 & 1980 & 500 & 1996 & 670 \\ \hline 1965 & 500 & 1981 & 420 & 1997 & 680 \\ \hline 1966 & 410 & 1982 & 640 & 1998 & 690 \\ \hline 1967 & 460 & 1983 & 770 & 1999 & 580 \\ \hline 1968 & 510 & 1984 & 710 & 2000 & 390 \\ \hline 1969 & 560 & 1985 & 680 & 2001 & 580 \\ \hline \end{array}$$

(c) Based on the shape of the distribution, does it seem reasonable to use $\bar{x}$ and s to describe the center and variability of this distribution? Why? Find $\bar{x}$ and s if you think they are a good choice. Otherwise, find the five-number summary.

Your bank account has a balance of $-\$ 12$. You deposit $\$ 60$. What is your new balance?

$(a)$ What is the direction and magnitude of an electric field that supports the weight of a free electron near the surface of Earth? $(b)$ Discuss what the small value for this field implies regarding the relative strength of the gravitational and electrostatic forces.

Mark Washington, CFA, is an analyst with BIC. One year ago, BIC analysts predicted that the U.S. equity market would most likely experience a slight downturn and suggested delta-hedging the BIC portfolio. As predicted, the U.S. equity markets did indeed experience a downturn of approximately 4% over a 12-month period. However, portfolio performance for BIC was disappointing, lagging its peer group by nearly 10%. Washington has been told to review the options strategy to determine why the hedged portfolio did not perform as expected. Washington considers a put option that has a delta of −.65. If the price of the underlying asset decreases by $6, then what is the best estimate of the change in option price?

The force $\mathbf{F}=30 \mathbf{i}+20 \mathbf{j}-10 \mathbf{k}\ (\mathrm{N})$ (a) Determine the magnitude of the moment of $\mathbf{F}$ about $A$. (b) Suppose that you can change the direction of $\mathbf{F}$ while keeping its magnitude constant, and you want to choose a direction that maximizes the moment of $\mathbf{F}$ about $A$. What is the magnitude of the resulting maximum moment?

An x-ray photon of initial frequency $3.0 \times 10 ^ { 19 }$ Hz collides with a free electron at rest; the scattered photon moves off at $90 ^ { \circ }$. What is the frequency of the scattered photon?

H.J. Heinz Company was founded in 1869 at Sharpsburg, Pennsylvania, by Henry J. Heinz. The company manufactures and markets food products throughout the world, including ketchup, condiments and sauces, frozen food, pet food, soups, and tuna. For two recent years, H.J. Heinz reported the following (in thousands):

$$\begin{array}{lrr} & \textbf{ Year 2 } & \textbf{ Year 1 } \\ \hline \text{Sales} & \$ 11,649,079 & \$ 10,706,588 \\ \text{Accounts receivable} & 993,510 & 1,265,032\\ \end{array}$$

Assume that the accounts receivable (in thousands) were $\$ 1,045,338$ at the beginning of Year 1.

b. Compute the days' sales in receivables at the end of Year 2 and Year 1. Use 365 days and round to one decimal place.

Given the following voltage transfer function:

$$H(s) = (Vo/Vi) = ((25 × 10^6)/(s² + 1000s + 25 × 10^6)).$$

a) At what frequencies (in radians per second) is the magnitude of the transfer function equal to unity? b) At what frequency is the magnitude of the transfer function maximum? c) What is the maximum value of the transfer function magnitude?

A semidefinite system has at least one_________ body motion.

Polo Ralph Lauren Corporation designs, markets, and distributes a variety of apparel, home decor, accessory, and fragrance products. The company's products include such brands as Polo by Ralph Lauren, Ralph Lauren Purple Label, Ralph Lauren, Polo Jeans Co., and Chaps. Polo Ralph Lauren reported the following (in thousands) for two recent years:

$$\begin{array}{lrr} & \textbf{For the Period Ending} \\ & \textbf{ Year 2 } & \textbf{Year 1} \\ \hline \text{Sales} & \$ 6,859,500 & \$ 5,660,300 \\ \text{Accounts receivable} & 690,000 & 592,700 \end{array}$$

Assume that accounts receivable (in millions) were $\$ 486,200$ at the beginning of Year 1.

c. What conclusions can be drawn from these analyses regarding Ralph Lauren's efficiency in collecting receivables?

Waldum Company purchased packaging equipment on January 5, 2012, for $135,000. The equipment was expected to have a useful life of three years, or 18,000 operating hours, and a residual value of$13,500. The equipment was used for 8,600 hours during 2012, 5,300 hours in 2013, and 4,100 hours in 2014.

Instructions

1. Determine the amount of depreciation expense for the years ended December 31, 2012, 2013, and 2014, by (a) the straight-line method, (b) the units-of-output method, and (c) the double-declining-balance method. Also determine the total depreciation expense for the three years by each method. The following columnar headings are suggested for recording the depreciation expense amounts:

$$\begin{array}{lcccc} \hspace{40pt}&\underline{\hspace{65pt}{\textbf{Depreciation Expense}\hspace{65pt}}}\\ \end{array}$$

$$\begin{array}{lcccc} {\textbf{}}&\textbf{Straight-}\hspace{15pt}&\textbf{Units-of-}\hspace{15pt}&\textbf{Double-Declining}\\ {\textbf{}}&\textbf{Line}\hspace{15pt}&\textbf{Output}\hspace{15pt}&\textbf{Balance}\\ {\textbf{Year}}\hspace{15pt}&\textbf{Method}\hspace{15pt}&\textbf{Method}\hspace{15pt}&\textbf{Method}\\\hline \end{array}$$

2. What method yields the highest depreciation expense for 2012?
3. What method yields the most depreciation over the three-year life of the equipment?

Consider the space X obtained from a seven-sided polygonal region by means of the labelling scheme $\text {abaaab}^{-1} a^{-1}$. Show that the fundamental group of X is the free product of two cyclic groups.