   Chapter 9.1, Problem 34E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

The numerical value of the volume of a cube equals the numerical value of its total surface area. What is the length of each edge of the cube?

To determine

To find:

The length of each edge of the cube such that the numerical value of the volume of the cube equals to the numerical value of its total surface area.

Explanation

The volume of any solid is the region of space occupied by it and in general the volume of a solid is given by the product of its base area and altitude.

For a regular solid prism with base area B and altitude h, the volume is given by the formula

V=Bh.

The total surface area (T) of a right rectangular prism with lateral area (L) and the base area (B) is given by

T=L+2B

Calculation:

Given,

Volume of the cube = Total surface area of the cube

Let ‘e’ be the length of each edge of the cube.

The volume of a right rectangular prism (cuboid) is given by the formula

V=lwh

But, here for the cube

l=e,

w=e, and

h=e

Thus,

V=e·e·e

V=e3

Now, the total surface area of a right rectangular prism is given by

T=L+2B

First let us calculate the lateral are of the prism (cube).

The lateral area (L) of a right rectangular prism with perimeter of the base (P) and the height ‘h’ is given by

L=Ph

Since, the prism is a cube we have all its edges are equal and is equal to ‘e

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