91Ó°ÊÓ

A manufacturer produces two models of elliptical cross-training exercise machines. The times for assembling, finishing, and packaging model \(\mathrm{X}\) are 3 hours, 3 hours, and 0.8 hour, respectively. The times for model Y are 4 hours, 2.5 hours, and 0.4 hour. The total times available for assembling, finishing, and packaging are 6000 hours, 4200 hours, and 950 hours, respectively. The profits per unit are \(300\) for model \(X\) and \(375\) for model Y. What is the optimal production level for each model? What is the optimal profit?

A humanitarian agency can use two models of vehicles for a refugee rescue mission. Each model A vehicle costs \(1000\) and each model B vehicle costs \(1500 .\) Mission strategies and objectives indicate the following constraints. " The agency must use a total of at least 20 vehicles. \(\cdot \mathrm{A}\) model \(\mathrm{A}\) vehicle can hold 45 boxes of supplies. \(\mathrm{A}\) model B vehicle can hold 30 boxes of supplies. The agency must deliver at least 690 boxes of supplies to the refugee camp. \(\cdot \mathrm{A}\) model \(\mathrm{A}\) vehicle can hold 20 refugees. A model \(\mathrm{B}\) vehicle can hold 32 refugees. The agency must rescue at least 520 refugees. What is the optimal number of vehicles of each model that should be used? What is the optimal cost?

Write the partial fraction decomposition of the improper rational expression. $$\frac{x^{2}-4 x}{x^{2}+x+6}$$

A company has budgeted a maximum of \(1,000,000\) for national advertising of an allergy medication. Each minute of television time costs \(100,000\) and each one-page newspaper ad costs \(20,000 .\) Each television ad is expected to be viewed by 20 million viewers, and each newspaper ad is expected to be seen by 5 million readers. The company's market research department recommends that at most \(80 \%\) of the advertising budget be spent on television ads. What is the optimal amount that should be spent on each type of ad? What is the optimal total audience?

Determine whether the statement is true or false. Justify your answer. When solving a linear programming problem, if the objective function has a maximum value at more than one vertex, then there are an infinite number of points that will produce the maximum value.

Rectangle: vertices at (4,3),(9,3),(9,9),(4,9)

A small software company invests \(\$ 16,000\) to produce a software package that will sell for \(\$ 55.95 .\) Each unit costs \(\$ 9.45\) to produce. (a) How many units must the company sell to break even? (b) How many units must the company sell to make a profit of \(\$ 100,000 ?\)

A mixture of 5 pounds of fertilizer \(A\), 13 pounds of fertilizer \(\mathrm{B},\) and 4 pounds of fertilizer \(\mathrm{C}\) provides the optimal nutrients for a plant. Commercial brand X contains equal parts of fertilizer \(\mathrm{B}\) and fertilizer \(\mathrm{C}\). Commercial brand Y contains one part of fertilizer \(\mathrm{A}\) and two parts of fertilizer B. Commercial brand Z contains two parts of fertilizer \(\mathrm{A}\), five parts of fertilizer \(\mathrm{B},\) and two parts of fertilizer C. How much of each fertilizer brand is needed to obtain the desired mixture?

A small corporation borrowed 775,000 dollar to expand its clothing line. Some of the money was borrowed at \(8 \%,\) some at \(9 \%,\) and some at \(10 \% .\) How much was borrowed at each rate when the annual interest owed was 67,500 dollar and the amount borrowed at \(8 \%\) was four times the amount borrowed at \(10 \% ?\)

(a) graph the systems representing the consumer surplus and producer surplus for the supply and demand equations and (b) find the consumer surplus and producer surplus. $$\begin{array}{cc}\text{Demand} && \text {Supply} \\ p=400-0.0002 x && p=225+0.0005 x \end{array}$$