A farmer is planning a rectangular area for her chickens. The area of the rectangle will be 800 square feet. Three sides of the rectangle will be formed by fencing, which costs $5 per foot. The fourth side of the rectangle will be formed by a portion of the barn wall, which requires no fencing. In order to minimize the cost of the fencing, how long should the fourth side be?

Let x represent the length in feet of the fourth side. Then the sides perpendicular to the barn wall will have length 800/x, and the total cost of the fence will be ...

cost = $5 × (x + 2·800/x)

The derivative of cost with respect to x will be zero when the cost is a minimum:

d(cost)/dx = 5 -8000/x^2 = 0

5x^2 = 8000 . . . . . multiply by x^2, add 8000

x = √1600 = 40 . . . . feet . . . . . . divide by 5, take the square root

The length of the fourth side should be 40 feet.

erinna erinna · Answer 2 · 2019-11-18T07:38:10+01:00

Answer:

40 feet.

Step-by-step explanation:

Area of rectangle = 800 square feet

Let the fourth side of the rectangle be x.

Length of the rectangle = x feet

The area of a rectangle is

[tex]A=length\times width[/tex]

[tex]800=x\times width[/tex]

[tex]\frac{800}{x}=width[/tex]

Cost of fencing = $5 per foot

Cost function on three sides is

[tex]C(x)=5[length+2(width)][/tex]

[tex]C(x)=5x+10(\frac{800}{x})[/tex]

[tex]C(x)=5x+\frac{8000}{x}[/tex]

Differentiate with respect to x.

[tex]C'(x)=5-\frac{8000}{x^2}[/tex] [tex][\because \frac{d}{dx}(\frac{1}{x})=-\frac{1}{x^2}][/tex]

To find the critical point equate C'(x)=0.

[tex]0=5-\frac{8000}{x^2}[/tex]

[tex]-5=-\frac{8000}{x^2}[/tex]

[tex]-5x^2=-8000[/tex]

Divide both sides by -5.

[tex]x^2=-\frac{8000}{-5}[/tex]

[tex]x^2=1600[/tex]

Taking square root on both sides,

[tex]x=40[/tex]

Differentiate C'(x) with respect to x.

[tex]C''(x)=\frac{16000}{x^3}>0[/tex]

Since C''(x)>0 for x=40, therefore cost of the fencing is minimum at x=40.

Thus, the measure of fourth side of the rectangle is 40 feet.