91Ó°ÊÓ

A soccer playing field of length \(x\) and width \(y\) has a perimeter of 360 meters. (a) Draw a rectangle that gives a visual representation of the problem. Use the specified variables to label the sides of the rectangle. (b) Show that the width of the rectangle is \(y=180-x\) and its area is \(A=x(180-x)\). (c) Use a graphing utility to graph the area equation. Be sure to adjust your window settings. (d) From the graph in part (c), estimate the dimensions of the rectangle that yield a maximum area. (e) Use your school's library, the Internet, or some other reference source to find the actual dimensions and area of a regulation Major League Soccer field and compare your findings with the results of part (d).

The table shows the life expectancies of a child (at birth) in the United States for selected years from 1930 through \(2000\) . $$\begin{array}{|c|c|}\hline \text { Year } & \text { Life Expectancy, \(y\) } \\\\\hline 1930 & 59.7 \\\1940 & 62.9 \\\1950 & 68.2 \\\1960 & 69.7 \\\1970 & 70.8 \\\1980 & 73.7 \\\1990 & 75.4 \\\2000 & 76.8 \\\\\hline\end{array}$$ A model for the life expectancy during this period is $$y=-0.002 t^{2}+0.50 t+46.6, \quad 30 \leq t \leq 100$$ where \(y\) represents the life expectancy and \(t\) is the time in years, with \(t=30\) corresponding to 1930. (a) Use a graphing utility to graph the data from the table and the model in the same viewing window. How well does the model fit the data? Explain. (b) Determine the life expectancy in 1990 both graphically and algebraically. (c) Use the graph to determine the year when life expectancy was approximately \(76.0 .\) Verify your answer algebraically. (d) One projection for the life expectancy of a child born in 2015 is \(78.9 .\) How does this compare with the projection given by the model? (e) Do you think this model can be used to predict the life expectancy of a child 50 years from now? Explain.

Determine whether the statement is true or false. Justify your answer. A function with a square root cannot have a domain that is the set of real numbers.

A pharmaceutical salesperson receives a monthly salary of 2500 dollar plus a commission of \(7 \%\) of sales. Write a linear equation for the salesperson's monthly wage \(W\) in terms of monthly sales \(S\).

If \(f\) is an even function, determine whether \(g\) is even, odd, or neither. Explain. (a) \(g(x)=-f(x)\) (b) \(g(x)=f(-x)\) (c) \(g(x)=f(x)-2\) (d) \(g(x)=f(x-2)\)

A roofing contractor purchases a shingle delivery truck with a shingle elevator for 42,000 dollar. The vehicle requires an average expenditure of 9.50 dollar per hour for fuel and maintenance, and the operator is paid 11.50 dollar per hour. (a) Write a linear equation giving the total cost \(C\) of operating this equipment for \(t\) hours. (Include the purchase cost of the equipment.) (b) Assuming that customers are charged 45 dollar per hour of machine use, write an equation for the revenue \(R\) derived from \(t\) hours of use. (c) Use the formula for profit \(P=R-C\) to write an equation for the profit derived from \(t\) hours of use. (d) Use the result of part (c) to find the break-even point - that is, the number of hours this equipment must be used to yield a profit of 0 dollars.

The length and width of a rectangular garden are 15 meters and 10 meters, respectively. A walkway of width \(x\) surrounds the garden. (a) Draw a diagram that gives a visual representation of the problem. (b) Write the equation for the perimeter \(y\) of the walkway in terms of \(x\) (c) Use a graphing utility to graph the equation for the perimeter. (d) Determine the slope of the graph in part (c). For each additional one- meter increase in the width of the walkway, determine the increase in its perimeter.