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Chapter 3 Solutions
College Algebra (MindTap Course List)
- Maximum Volume You construct an open box with locking tabs from a square piece of material, 24 inches on a side, by cutting equal sections from the corners and folding along the dashed lines (see figure). (a) Write a function V that represents the volume of the box. (b) Determine the domain of the function V . (c) Sketch a graph of the function and estimate the value of x for which V(x) is a maximum.arrow_forwardMaximum Volume You construct an open box from a square piece of material, 36 inches on a side, by cutting equal squares with sides of length x from the corners and turning up the sides (see figure). (a) Write a function V that represents the volume of the box. (b) Determine the domain of the function V . (c) Use a graphing utility to construct a table that shows the box heights x and the corresponding volumes V(x) . Use the table to estimate the dimensions that produce a maximum volume. (d) Use the graphing utility to graph V and use the graph to estimate the value of x for which V(x) is a maximum. Compare your result with that of part (c).arrow_forwardGraphical Reasoning Sketch the graph of the function f(x)=x4.Explain how the graph of each function gdiffers (if it does) from the graph of f.Determine whether gis even, odd, or neither. ag(x)=f(x)+2bg(x)=f(x+2)cg(x)=f(x)dg(x)=f(x)eg(x)=f12xfg(x)=12f(x)gg(x)=f(x3/4)hg(x)=(ff)(x)arrow_forward
- Geometry You want to make an open box from a rectangular piece of material, 15 centimeters by 9 centimeters, by cutting equal squares from the corners and turning up the sides. (a) Let x represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume V of the box as a function of x. Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that yield a maximum volume. (d) Find values of x such that V=56. Which of these values is a physical impossibility in the construction of the box? Explain.arrow_forwardVolume of a box A cardboard box has a square base, with each edge of the base having length x inches, as shown in the figure. The total length of all 12 edges of the box is 144 in. (a) Show that the volume of the box is given by the function V(x)=2x2(18x). (b) What is the domain of V? Use the fact that length and volume must be positive. (c) Draw a graph of the function V and use it to estimate the maxim volume for such a box.arrow_forwardSolving Inequalities Solve the inequalities in Exercises S-5 through S-19. xx2+3arrow_forward
- Profit A company that produces calculators estimates that the profit P(in dollars) from selling a specific model of calculator is given by P=152x3+7545x2169,625,0x45 where xis the advertising expense (in tens of thousands of dollars).For this model of calculator, an advertising expense of $400,000(x=40)results in a profit of $2,174,375. (a) Use a graphing utility to graph the profit function. (b) Use the graph from part (a) to estimate another amount the company can spend on advertising that results in the same profit. (c) Use synthetic division to confirm the result of part (b) algebraically.arrow_forwardGeometry A rectangular package to be sent by a delivery service (see figure) has a combined length and girth (perimeter of a cross section) of 120 inches. (a) Use the diagram to write the volume V of the package as a function of x. (b) Use a graphing utility to graph the function and approximate the dimensions of the package that yield a maximum volume. (c) Find values of x such that V=13,500. Which of these values is a physical impossibility in the construction of the package? Explain.arrow_forwardPolynomial Inequalities Solve the inequality. (x1)(x+2)(x3)(x+4)0arrow_forward
- Average Speed A driver’s average speed is 50 miles per hour on a round trip between two cities 100 miles apart. The average speeds for going and returning were xand ymiles per hour, respectively. (a) Show that y=25xx25. (b) Determine the vertical and horizontal asymptotes of the graph of the function. (c) Use a graphing utility to graph the function. (d) Complete the table. (e) Are the results in the table what you expected? Explain. (f ) Is it possible to average 20 miles per hour in one direction and still average 50 miles per hour on the round trip? Explain.arrow_forwardSelf Check Fill in the blanks: The graph of g(x)=x2+3is identical to the graph of fx=x2, except that it is translated _______ units _______. The graph of hx=x2-4 is identical to the graph of fx=x2, except that it is translated _______ units __________.arrow_forwardSolving Inequalities Solve the inequalities in Exercises S-5 through S-19. x2+4x Be careful. The graphs do not cross.arrow_forward
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