Chapter 2: Q. 26 (page 121)

An open box with a square base is required to have a volume of 10 cubic feet.
(a) Express the amount A of material used to make such a
box as a function of the length x of a side of the square
base.
(b) How much material is required for a base 1 foot by
1 foot?
(c) How much material is required for a base 2 feet by
2feet?
(d) Use a graphing utility to graph $A = A (x)$ . For what value
of x is A smallest?

Expert verified

Part (a) Area of material used for constructing open box is $A (x) = x^{2} + \frac{40}{x}$ .

Part (b) $A (1) = 41 i n^{2}$

Part (c) $A (2) = 24 i n^{2}$

Part (d) A(x) is smallest whenx is $\sqrt[3]{20}$ feet.

Step by step solution

Volume of the open box is

$V = x^{2} h 10 = x^{2} h \frac{10}{x^{2}} = h$

Area of base $+ 4$ (area of sides)

$A (x) = x^{2} + 4 x h A (x) = x^{2} + 4 x (\frac{10}{x^{2}}) A (x) = x^{2} + \frac{40}{x}$

This gives

$A (1) = 1^{2} + \frac{40}{1} A (1) = 41 {in}^{2}$

This gives

$A (2) = 2^{2} + \frac{40}{2} A (2) = 4 + 20 A (2) = 24 {in}^{2}$

From the graph,

A(x) is smallest when $x = 2.71 = \sqrt[3]{20}$ feet.

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