Under the last-in first-out (LIFO) inventory valuation method, a price index for inventory must be established for tax purposes. The quantity weights are based on year-ending inventory levels. Use the beginning-of-the-year price per unit as the base-period price and develop a weighted aggregate index for the total inventory value at the end of the year. What type of weighted aggregate price index must be developed for the LIFO inventory valuation?

$$\begin{matrix} \text{ } & \text{Ending} & \text{Unit} & \text{Price (\$)}\\ \hline \text{Product} & \text{Inventory} & \text{Beginning} & \text{Ending}\\ \hline \text{A} & \text{500} & \text{.15} & \text{.19}\\ \text{B} & \text{50} & \text{1.60} & \text{1.80}\\ \text{C} & \text{100} & \text{4.50} & \text{4.20}\\ \text{D} & \text{40} & \text{12.00} & \text{13.20}\\ \end{matrix}$$

Determine the measure of angle A in triangle ABC, where A(1, 1, 8), B(4, −3, −4), and C(−3, 1, 5). Express your answer in degrees rounded to two decimal places.

The scale of a map Is

$$2 \mathrm { cm } : 15 \mathrm { km }$$

Find the actual distance for each map distance.

$$17.4 \mathrm { cm }$$

Electric power consumption is measured in kilowatt-hours (kWh). The local utility company offers an interrupt program whereby commercial customers that participate receive favorable rates but must agree to cut back consumption if the utility requests them to do so. Timko Products has agreed to cut back consumption from noon to 8:00 P.M. on Thursday. To determine Timko’s savings, the utility must estimate Timko’s normal power usage for this period of time. Data on Timko’s electric power consumption for the previous 72 hours are shown below.

$$\begin{matrix} \text{Time Period} & \text{Monday} & \text{Tuesday} & \text{Wednesday} & \text{Thursday}\\ \hline \text{12–4 A.M.} & \text{—} & \text{19,281} & \text{31,209} & \text{27,330}\\ \text{4–8 A.M.} & \text{—} & \text{33,195} & \text{37,014} & \text{32,715}\\ \text{8–12 noon} & \text{—} & \text{99,516} & \text{119,968} & \text{152,465}\\ \text{12–4 P.M.} & \text{124,299} & \text{123,666} & \text{156,033}\\ \text{4–8 P.M.} & \text{113,545} & \text{111,717} & \text{128,889}\\ \text{8–12 midnight} & \text{41,300} & \text{48,112} & \text{73,923}\\ \end{matrix}$$

a. Construct a time series plot. What type of pattern exists in the data? b. Use the following dummy variables to develop an estimated regression equation to account for any seasonal effects in the data. Time1 = 1 for time period 12–4 A.M.; 0 otherwise Time2 = 1 for time period 4–8 A.M; 0 otherwise Time3 = 1 for time period 8–12 noon; 0 otherwise Time4 = 1 for time period 12–4 P.M; 0 otherwise Time5 = 1 for time period 4–8 P.M; 0 otherwise. c. Use the estimated regression equation developed in part (b) to estimate Timko’s normal usage over the period of interrupted service. d. Let Period = 1 to refer to the observation for Monday in the time period 12–4 P.M.; Period = 2 to refer to the observation for Monday in the time period 4–8 P.M; . . . and Period =18 to refer to the observation for Thursday in the time period 8–12 noon. Using the dummy variables defined in part (b) and Period, develop an estimated regression equation to account for seasonal effects and any linear trend in the time series. e. Using the estimated regression equation developed in part (d), estimate Timko’s normal usage over the period of interrupted service.

Evaluate the integral by making a substitution that converts the integrand to a rational function.

$$\int \frac { e ^ { 3 x } } { e ^ { 2 x } + 4 } d x$$

The two partners in a partnership share the profits of their business in the ratio 5:3. Last year the profits were $360,000. Find the amount received by each partner.

A circle is centered at the origin and has a radius of 1. a. What is the equation of the circle? b. If $x=\frac{1}{2}$ in the equation you wrote in part (a), what is the exact value of y? You should get two values.

Sketch and estimate the area determined by the intersections of the curves. $y=e^{x}, y=1-x^{2}$

At 7 a.m., eight people leave their cars for repair at Harry’s Auto Repair Shop. If person i’s car is ready by time t (7 a.m. = time 0, and so on), he will pay Harry ri(t) dollars. For example, if person 2’s car must be ready by 2 p.m., we may have r2(8) = 0. Harry estimates that with probability pi(t), it will take t hours to repair person i’s car. Formulate a dynamic programming recursion that will enable Harry to maximize his expected revenue for the day. His workday ends at 5 p.m. = time 10.

Use the divisibility test to determine if 2345 is divisible by 4. Explain your answer.

Use a graphing utility to graph the function and identify any horizontal asymptotes. f(x) = √(9x²-2) / (2x+1)

The Goodman Tire and Rubber Company periodically tests its tires for tread wear under simulated road conditions. To study and control the manufacturing process, 20 samples, each containing three radial tires, were chosen from different shifts over several days of operation, with the following results. Assuming that these data were collected when the manufacturing process was believed to be operating in control, develop the R and $\overline x$ charts.

$$\begin{matrix} \text{Sample} & \text{ } & \text{Tread Wear*}\\ \text{1} & \text{31} & \text{42} & \text{28}\\ \text{2} & \text{26} & \text{18} & \text{35}\\ \text{3} & \text{25} & \text{30} & \text{34}\\ \text{4} & \text{17} & \text{25} & \text{21}\\ \text{5} & \text{38} & \text{29} & \text{35}\\ \text{6} & \text{41} & \text{42} & \text{36}\\ \text{7} & \text{21} & \text{17} & \text{29}\\ \text{8} & \text{32} & \text{26} & \text{28}\\ \text{9} & \text{41} & \text{34} & \text{33}\\ \text{10} & \text{29} & \text{17} & \text{30}\\ \text{11} & \text{26} & \text{31} & \text{40}\\ \text{12} & \text{23} & \text{19} & \text{25}\\ \text{13} & \text{17} & \text{24} & \text{32}\\ \text{14} & \text{43} & \text{35} & \text{17}\\ \text{15} & \text{18} & \text{25} & \text{29}\\ \text{16} & \text{30} & \text{42} & \text{31}\\ \text{17} & \text{28} & \text{36} & \text{32}\\ \text{18} & \text{40} & \text{29} & \text{31}\\ \text{19} & \text{18} & \text{29} & \text{28}\\ \text{20} & \text{22} & \text{34} & \text{26}\\\\ *\text{Hundredths of an inch} \end{matrix}$$

Determine by inspection a subset of $S$ containing the fewest vectors that has the same span as $S$.

$$\begin{equation*}\begin{Bmatrix}\begin{bmatrix}1\\2\\-3\end{bmatrix},\begin{bmatrix}4\\-6\\2\end{bmatrix},\begin{bmatrix}-2\\3\\-1\end{bmatrix} ,\begin{bmatrix}-3\\-6\\9\end{bmatrix}\end{Bmatrix}\end{equation*}$$

Twenty-five samples of 100 items each were inspected when a process was considered to be operating satisfactorily. In the 25 samples, a total of 135 items were found to be defective. a. What is an estimate of the proportion defective when the process is in control? b. What is the standard error of the proportion if samples of size 100 will be used for statistical process control? c. Compute the upper and lower control limits for the control chart.