# The function of the surface area of the open rectangular box with a square base in terms of the length of a side of the base and the domain of the function. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 1.1, Problem 57E
To determine

## To find: The function of the surface area of the open rectangular box with a square base in terms of the length of a side of the base and the domain of the function.

Expert Solution

The formula for the surface area of the open rectangular box as a function of the length of a side of the base is S(a)=a2+8a .

The domain of the surface area of the open rectangular box as a function of the length of a side of the base is a>0 .

### Explanation of Solution

Given:

The volume of the open rectangular box with a square base is 2 m3.

Formula used:

Volume of the rectangular box of length l, width b, height h is, V=l×b×h .

Surface area of the open rectangular box of length l, width b, height h is, S=2(lh+wh)+lb .

Calculation:

Let the length of the open rectangular box be a and the height of the open rectangular box is h.

Since the base of the open rectangular box is a square, the width of the box is same as the length. Therefore, the width of the open rectangular box is a.

Volume of the open rectangular box is, V=a×a×h .

That is, V=a2h

Since the volume of the open rectangular box with a square base is 2 m3, the height of the box is computed as follows:

2=a2hh=2a2

Thus, the surface area of the open rectangular box is S=2(ah+ah)+a2 .

That is, S=4ah+a2

Substitute the value of h in S=4ah+a2 and simplify.

S=4a(2a2)+a2=8a+a2

Therefore, the formula for the surface area (S) of the open rectangular box as a function of the length of a side of the base (a) is S(a)=a2+8a .

The domain of the surface area of the open rectangular box as a function of the length of a side of the base is a>0 .

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