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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?

Short Answer

Expert verified

Part (a) Area of material used for constructing open box is A(x)=x2+40x.

Part (b) A(1)=41in2

Part (c) A(2)=24in2

Part (d) A(x) is smallest whenx is203feet.

Step by step solution

01

Part (a). Step 1. Use the given volume of the open box.

Volume of the open box is

V=x2h10=x2h10x2=h

02

Part (a) Step 2. The area of the material required to make the open box is

Area of base +4(area of sides)

A(x)=x2+4xhA(x)=x2+4x(10x2)A(x)=x2+40x

03

Part (b) Step 1. Substitute x=1 inch in A(x)=x2+40x.

This gives

A(1)=12+401A(1)=41in2

04

Part (c) Step 1. Substitute x=2 inches in A(x)=x2+40x.

This gives

A(2)=22+402A(2)=4+20A(2)=24in2

05

Part (d) Step 1. The graph of A(x)=x2+40x is

From the graph,

A(x) is smallest whenx=2.71=203feet.

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