# The function that models the area of corral in terms of the width x of the corral.

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 3.1, Problem 75E

(a)

To determine

## To evaluate: The function that models the area of corral in terms of the width x of the corral.

Expert Solution

The function that models area of corral is A(x)=1200xx2 .

### Explanation of Solution

Given:

Corral has 2400 ft of fencing to fence in a rectangular horse corral.

Calculation:

The width of the corral is x .

Let the length of the corral is y .

The parameter of the rectangular corral is,

2(x+y)=2400x+y=1200 (1)

From the equation (1),

y=1200x (2)

The area of the corral is,

A=Length×width=x×y (3)

From the equation (2) and (3),

A(x)=x×(1200x)=1200xx2 (4)

The area of the corral in the term of the width x is,

A(x)=1200xx2

Thus, the function that models area of corral is A(x)=1200xx2 .

(b)

To determine

### To evaluate: The dimension of corral to maximize area corral.

Expert Solution

The length and width of the corral is 600ft and 600ft respectively.

### Explanation of Solution

Given:

Corral has 2400 ft of fencing to fence in a rectangular horse corral.

Calculation:

The width of the corral is x .

Let the length of the corral is y .

The parameter of the rectangular corral is,

2(x+y)=2400x+y=1200 (1)

From the equation (1),

y=1200x (2)

The area of the corral is,

A=Length×width=x×y (3)

From the equation (2) and (3),

A(x)=x×(1200x)=1200xx2 (4)

The area of the corral in the term of the width x is,

A(x)=x2+1200x (5)

The standard form of function,

f(x)=ax2+bx+c (6)

The maximum or minimum value of the function occurs at,

x=b2a (7)

If a>0 , then the minimum value is f(b2a) .

If a<0 , then the maximum value is f(b2a) .

From the equation (5) and (6),

a=1b=1200

Substitute 1 for a and 1200 for b in the equation (7),

x=12002×(1)=600

The width of the corral is 600 ft.

Substitute 600 for x in the equation (2),

y=1200600=600

The length of the corral is 600 ft.

The function has maximum value,

a=1<0 .

Thus, the length and width of the corral is 600ft and 600ft respectively.

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