91Ó°ÊÓ

Supply a valid technol\(\operatorname{og} y\) formula and then use technology to compute the missing values in the following table: HINT [See Quick Examples on page 633.] $$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & & \\ \hline \end{array} $$ \(h(x)=3.42\left(3^{-x / 5}\right)\)

Convert the given exponential function to the form indicated. Round all coefficients to four significant digits. $$ f(x)=2.1 e^{-0.1 x} ; f(x)=A b^{x} $$

How long will it take a \(\$ 500\) investment to be worth \(\$ 700\) if it is continuously compounded at \(15 \%\) per year? (Give the answer to two decimal places.) HINT [See Example 3.]

T College Basketball: Women The following table shows the number of NCAA women's college basketball teams in the United States for various years since \(1990 .{ }^{47}\) \begin{tabular}{|r|c|c|c|c|c|c|} \hline \(\boldsymbol{t}\) (year since 1990) & 0 & 5 & 10 & 11 & 12 & 13 \\ \hline Teams & 1,549 & 1,732 & 1,888 & 1,895 & 1,911 & 1,976 \\ \hline \(\boldsymbol{t}\) (year since 1990) & 14 & 15 & 16 & 17 & 18 & \\ \hline Teams & 1,989 & 2,019 & 2,002 & 2,040 & 2,060 & \\ \hline \end{tabular} a. What is the logistic regression model for the data? (Round all parameters to three significant digits.) At what value does the model predict that the number of basketball teams will level off? b. According to the model, for what value of \(t\) is the regression curve steepest? Interpret the answer. c. Interpret the coefficient \(b\) in the context of the number of women's basketball teams.

The latest demand equation for your Yoda vs. Alien T-shirts is given by $$ q=-40 x+600 $$ where \(q\) is the number of shirts you can sell in one week if you charge $$\$ x$$ per shirt. The Student Council charges you $$\$ 400$$ per week for use of their facilities, and the T-shirts cost you $$\$ 5$$ each. Find the weekly cost as a function of the unit price \(x\). Hence, find the weekly profit as a function of \(x\) and determine the unit price you 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: $$ \begin{array}{ll} \text { Fixed costs: } & \$ 120,000 \text { per month } \\ \text { Variable costs: } & \$ 80,000 \text { per contract } \end{array} $$ 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.

I would like my investment to double in value every 3 years. At what rate of interest would I need to invest it, assuming the interest is compounded continuously? HIIII [See Duick Examnles nane 655.]

Logistic functions are commonly used to model the spread of epidemics. Given this fact, explain why a logistic function is also useful to model the spread of a new technology.

My investment in OHaganBooks.com stocks is losing half its value every 2 years. Find and interpret the associated decay rate. HINT [See Quick Examples page 655.]

Why is a logistic function more appropriate than an exponential function for modeling the spread of an epidemic?