91Ó°ÊÓ

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{3}-x^{2}+x+39$$

A page that is \(x\) inches wide and \(y\) inches high contains 30 square inches of print. The top and bottom margins are each 1 inch deep, and the margins on each side are 2 inches wide (see figure). (a) Write a function for the total area \(A\) of the page in terms of \(x\) (b) Determine the domain of the function based on the physical constraints of the problem. (c) Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper is used.

The path of a diver is given by the function $$f(x)=-\frac{4}{9} x^{2}+\frac{24}{9} x+12$$ where \(f(x)\) is the height (in feet) and \(x\) is the horizontal distance from the end of the diving board (in feet). What is the maximum height of the diver?

A rectangular parking lot with a perimeter of 440 feet is to have an area of at least 8000 square feet. Within what bounds must the length of the rectangle lie?

The path of a punted football is given by the function $$f(x)=-\frac{16}{2025} x^{2}+\frac{9}{5} x+1.5$$ where \(f(x)\) is the height (in feet) and \(x\) is the horizontal distance (in feet) from the point at which the ball is punted. (a) How high is the ball when it is punted? (b) What is the maximum height of the punt? (c) How long is the punt?

Determine whether the statement is true or false. Justify your answer. The graph of a rational function can never cross one of its asymptotes.

Determine whether the statement is true or false. Justify your answer. The graph of a rational function can have a vertical asymptote, a horizontal asymptote, and a slant asymptote.

The total revenue \(R\) earned (in thousands of dollars) from manufacturing handheld video games is given by $$R(p)=-25 p^{2}+1200 p$$ where \(p\) is the price per unit (in dollars). (a) Find the revenues when the prices per unit are \(\$ 20\) \(\$ 25,\) and \(\$ 30\) (b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results.

When two resistors of resistances \(R_{1}\) and \(R_{2}\) are connected in parallel (see figure), the total resistance \(R\) satisfies the equation \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\) Find \(R_{1}\) for a parallel circuit in which \(R_{2}=2\) ohms and \(R\) must be at least 1 ohm.

An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200 -meter single-lane running track. (a) Draw a diagram that gives a visual representation of the problem. Let \(x\) and \(y\) represent the length and width of the rectangular region, respectively. (b) Determine the radius of each semicircular end of the room. Determine the distance, in terms of \(y,\) around the inside edge of each semicircular part of the track. (c) Use the result of part (b) to write an equation, in terms of \(x\) and \(y,\) for the distance traveled in one lap around the track. Solve for \(y\) (d) Use the result of part (c) to write the area \(A\) of the rectangular region as a function of \(x .\) What dimensions will produce a rectangle of maximum area?