If the revenue function for a particular commodity is $R(p)=-50 p^{2}+60 p$, what is the (linear) demand function? Give a reason for your answer.

Write each expression as a single logarithm. $2 \log _{2}(x+1)-\log _{2}(x+3)-\log _{2}(x-1)$

Use the point-slope formula to write an equation of the line having the given conditions. Write the answer in slope-intercept form (if possible). Given a line defined by y=-2, what is the slope of the line?

A sporting goods store finds that if it sells x exercise machines per day, its costs will be $C(x)=100 x+3200$ and its revenue will be $R(x)=-2 x^{2}+300 x$ (both in dollars). a. Find the store’s break-even points. b. Find the number of sales that will maximize profit, and the maximum profit.

Two fraternities, Sig Ep and Ep Sig, plan to raise money jointly to benefit homeless people on Long Island. They will sell Yoda vs. Alien T-shirts in the student center, but are not sure how much to charge. Sig Ep treasurer Augustus recalls that they once sold 400 shirts in a week at $8 per shirt, but Ep Sig treasurer Julius has solid research indicating that it is possible to sell 600 per week at$4 per shirt. a. Based on this information, construct a linear demand equation for Yoda vs. Alien T-shirts, and hence obtain the weekly revenue R as a function of the unit price x. b. The university administration charges the fraternities a weekly fee of $500 for use of the Student Center. Write down the monthly profit P as a function of the unit price x, and hence determine how much the fraternities should charge to obtain the largest possible weekly profit. What is the largest possible weekly profit?

As sales manager for Montevideo Productions, Inc., you are planning to review the prices you charge clients for television advertisement development. You currently charge each client an hourly development fee of $2,500. With this pricing structure, the demand, measured by the number of contracts Montevideo signs per month, is 15 contracts. This is down 5 contracts from the figure last year, when your company charged only$2,000. a. Construct a linear demand equation giving the number of contracts q as a function of the hourly fee p Montevideo charges for development. b. On average, Montevideo bills for 50 hours of production time on each contract. Give a formula for the total revenue obtained by charging $p per hour. c. The costs to Montevideo Productions are estimated as follows:$

$$\begin{matrix} \text{Fixed costs:} & \text{\$120,000 per month}\\ \text{Variable costs:} & \text{\$80,000 per contract}\\ \end{matrix}$$

$Express Montevideo Productions’ monthly cost (i) as a function of the number q of contracts and (ii) as a function of the hourly production charge p. d. Express Montevideo Productions’ monthly profit as a function of the hourly development fee p and hence the price it should charge to maximize the profit.

Find the midpoint of the line segment with endpoints at the given coordinates.

$$(8, 3), (-4, 9)$$

Choose the vocabulary word that answers each riddle. I refer to things that do not happen in daylight.

The demand for the diet pills is $50$ pills at a price of $\$47.50$ and $80$ pills at a price of $\$32.50$. Determine a linear demand function for these pills.

You operate a gaming Web site, www.mudbeast.net, where users must pay a small fee to log on. When you charged $2 the demand was 280 log-ons per month. When you lowered the price to$1.50, the demand increased to 560 logons per month. a. Construct a linear demand function for your Web site and hence obtain the monthly revenue R as a function of the log-on fee x. b. Your Internet provider charges you a monthly fee of $30 to maintain your site. Express your monthly profit P as a function of the log-on fee x, and hence determine the logon fee you should charge to obtain the largest possible monthly profit. What is the largest possible monthly profit?

The market research department of the Better Baby Buggy Co. predicts that the demand equation for its buggies is given by q=−0.5p+140, where q is the number of buggies it can sell in a month if the price is $p per buggy. At what price should it sell the buggies to get the largest revenue? What is the largest monthly revenue?

The table shows the retail market share of passenger cars from Ford Motor Company as a percentage of the U.S. market.

$$\begin{matrix} \text{Year } & \text{Market Share}\\ \text{1980} & \text{}{17.2 \%}\\ \text{1985} & \text{}{18.8 \%}\\ \text{1990} & \text{}{20.0 \%}\\ \text{1995} & \text{}{20.7 \%}\\ \text{2000} & \text{}{20.2 \%}\\ \text{2005} & \text{}{17.4 \%}\\ \text{2010} & \text{}{16.4 \%}\\ \end{matrix}$$

A mathematical model for this data is given by $f(x)=-0.0169 x^{2}+0.47 x+17.1$ where x=0 corresponds to 1980 (A) Complete the following table. Round values of f(x) to one decimal place.

$$\begin{matrix} \text{x} & \text{Market Share} & \text{f(x)}\\ \text{0} & \text{17.2}\\ \text{5} & \text{18.8}\\ \text{10} & \text{20.0}\\ \text{15} & \text{20.7}\\ \text{20} & \text{20.2}\\ \text{25} & \text{17.4}\\ \text{30} & \text{16.4}\\ \end{matrix}$$

(B) Sketch the graph of f and the market share data in the same coordinate system. (C) Use values of the modeling function f to estimate Ford's market share in 2020 and in 2025. (D) Write a brief description of Ford's market share from 1980 to 2010.

Solve the equation for x. fx - gx = h

Use synthetic division to perform the division. Divide

$$8 - 3 x + 7 x ^ { 2 } + 2 x ^ { 3 }$$

by x + 5.