375 3.4 Polynomial Functions: Graphs, Applications, and Models (Modeling) The following exercises are geometric in nature and lead to polynomial models. Solve each problem. 91. Volume of a Box A rectangular piece of cardboard measuring 12 in. by 18 in. is to be made into a box with an open top by cutting equal-size squares from each corner and folding up the sides. Let x represent the length of a side of each such square in inches. Give approximations to the nearest hundredth. 12 in. 18 in. x x x x x x x x (a) Give the restrictions on x. (b) Determine a function V that gives the volume of the box as a function of x. (c) For what value of x will the volume be a maximum? What is this maximum volume? (Hint: Use the function of a graphing calculator that allows us to determine a maximum point within a given interval.) (d) For what values of x will the volume be greater than 80 in.3? 92. Construction of a Rain Gutter A piece of rectangular sheet metal is 20 in. wide. It is to be made into a rain gutter by turning up the edges to form parallel sides. Let x represent the length of each of the parallel sides. Give approximations to the nearest hundredth. x x 20 – 2x 20 in. (a) Give the restrictions on x. (b) Determine a function that gives the area of a cross section of the gutter. (c) For what value of x will be a maximum (and thus maximize the amount of water that the gutter will hold)? What is this maximum area? (d) For what values of x will the area of a cross section be less than 40 in.2? 93. Sides of a Right Triangle A certain right triangle has area 84 in.2. One leg of the triangle measures 1 in. less than the hypotenuse. Let x represent the length of the hypotenuse. a 5 84 in.2 x (a) Express the length of the leg mentioned above in terms of x. Give the domain of x. (b) Express the length of the other leg in terms of x. (c) Write an equation based on the information determined thus far. Square both sides and then write the equation with one side as a polynomial with integer coefficients, in descending powers, and the other side equal to 0. (d) Solve the equation in part (c) graphically. Find the lengths of the three sides of the triangle.
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