Concept explainers
Maximum Volume You construct an open box from a square piece of material,
(a) Write a function
(b) Determine the domain of the function
(c) Use a graphing utility to construct a table that shows the box heights
(d) Use the graphing utility to graph
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Chapter 3 Solutions
College Algebra
- Volume of a Box An open box is to constructed from a piece of cardboard 20 cm by 40 cm by cutting squares of side length x from each corner and folding up the sides, as shownin figure. (a) Express the volume V of the box as a function of x. (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 maximum volume for such a box.arrow_forwardGeometry 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_forwardMaximum 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_forward
- Revenue A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function R(x)=80x0.4x2, where the revenue R(x) is measured in dollars. What is the maximum revenue? and how many units should be manufactured to obtain this maximum?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_forwardMaximum Area A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). The perimeter of the window is 16 feet. (a) Write the area A of the window as a function of x. (b) What dimensions produce a window of maximum area?arrow_forward
- Geometry 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_forwardGeometry The perimeter of a rectangle is 200meters. (a) Draw a diagram that gives a visual representation of the problem. Let x and y represent the length and width of the rectangle, respectively. (b) Write yas a function of x.Use the result to write the area A as a function of x. (c) Of all possible rectangles with perimeters of 200meters, find the dimensions of the one with the maximum area.arrow_forwardConcentration of a Mixture A 1000-liter tank contains 50 liters of a 25brine solution. You add xliters of a 75brine solution to the tank. (a) Show that the concentration C, the proportion of brine to total solution, in the final mixture is C=3x+504(x+50). (b) Determine the domain of the function based on the physical constraints of the problem. (c) Sketch the graph of the concentration function. (d) As the tank is filled, what happens to the rate at which the concentration of brine is increasing? What percent does the concentration of brine appear to approach?arrow_forward
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