Minimizing Surface Area United Parcel Service has contracted you to design a closed box with a square base that has a volume of 10,000 cubic inches. See the illustration. (a) Express the surface area S of the box as a function of x . (b) Using a graphing utility, graph the function found in part (a). (c) What is the minimum amount of cardboard that can be used to construct the box? (d) What are the dimensions of the box that minimize the surface area? (e) Why might UPS be interested in designing a box that minimizes the surface area?
Minimizing Surface Area United Parcel Service has contracted you to design a closed box with a square base that has a volume of 10,000 cubic inches. See the illustration. (a) Express the surface area S of the box as a function of x . (b) Using a graphing utility, graph the function found in part (a). (c) What is the minimum amount of cardboard that can be used to construct the box? (d) What are the dimensions of the box that minimize the surface area? (e) Why might UPS be interested in designing a box that minimizes the surface area?
Solution Summary: The author explains that UPS would like to design a box that has the largest volume with the least amount of surface area.
Minimizing Surface Area United Parcel Service has contracted you to design a closed box with a square base that has a volume of 10,000 cubic inches. See the illustration.
(a) Express the surface area
of the box as a function of
.
(b) Using a graphing utility, graph the function found in part (a).
(c) What is the minimum amount of cardboard that can be used to construct the box?
(d) What are the dimensions of the box that minimize the surface area?
(e) Why might UPS be interested in designing a box that minimizes the surface area?
Answer:
A carpenter wants to make a rectangular bookcase like
the one shown here. It should have a horizontal top, bottom, and
middle shelf, and vertical back and two sides. The depth of the
bookcase must be 1 ft, and its total volume must be 12 ft.
a. Express the total surface area of the bookcase as a function of
its height.
Answer:
b. What height and width of the bookcase will minimize its total surface area?
Chapter 4 Solutions
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