Question

A fence is to be built to enclose a rectangular area of "200" square feet. The fence along three sides is to be made of material that costs "5" dollars per foot, and the material for the fourth side costs 15 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.

Answer 1

Answer:

20 ft by 10 ft.

Step-by-step explanation:

Let the dimension of the rectangular enclosure be x ft by y ft

• Area to be enclosed $=200 ft^2$
• Therefore: Area=xy=200

Perimeter, P(x,y)=2(x+y)=x+x+2y

Fencing material along three sides costs $5 per foot

Fencing material for the fourth side costs $15 per foot.

Therefore, Cost of Fencing=$(15x+5x+5y+5y)

C(x,y)=20x+10y

We can write the cost function as a function of one variable by substituting for y.

Recall: xy=200

$y=\dfrac{200}{x}$

Therefore:

$C(x)=20x+10(\dfrac{200}{x})\\C(x)=\dfrac{20x^2+2000}{x}$

To determine dimensions of the enclosure that is most economical to construct, we minimize C(x) by taking its derivative and solving for its critical points.

$C'(x)=\dfrac{20x^2-2000}{x^2}\\ When C'(x)=0\\20x^2-2000=0\\20x^2=2000\\x^2=100\\x^2=10^2\\x=10 ft$

Recall that:

$y=\dfrac{200}{x}\\y=\dfrac{200}{10}\\y=20 ft$

Therefore, the dimension of the enclosure that is most economical to construct is 20 ft by 10 ft.

• Where the side length of 20 ft is made of materials that costs $5 per foot
• One of the side of length 10 ft is made of materials that costs $5 per foot
• The other side of length 10 ft is made of materials that costs $15 per foot