Gabriel wants to build a rectangular pen for his animals. One side of the pen will be against the barn; the other three sides will be enclosed with wire fencing. If Gabriel has 650 feet of fencing
A) What dimensions would maximize the area of the pen?
B) Let w be the width of the enclosure (perpendicular to the barn) and let l be the length of the enclosure (parallel to the barn). Write an function for the area A of the enclosure in terms of w.
C) What width w would maximize the area?
D) What is the maximum area?

(a) 162.5 ft by 325 ft.

(b)

(c)Width = 162.5 feet

(d)

Step-by-step explanation:

Let w be the width of the enclosure (perpendicular to the barn)

Let l be the length of the enclosure (parallel to the barn).

Perimeter of the fence = 650 feet

Since one side will be against the barn and we are fencing only three sides,

Perimeter = 2w+l

Therefore:

2w+l =650

Area of the pen A(l,w)=lw

Substituting l=650-2w into A(l,w)

The dimension that will maximize the area of the pen occurs when the derivative of A(w)=0. (i.e. at the maximum point)

650-4w=0

4w=650

w=162.5 feet

Recall: l=650-2w

l=650 -2(162.5)=650-325=325 feet

l=325 feet

(a)The dimensions that will maximize the area of the pen is 162.5 ft by 325 ft.

(b)

If w =the width of the enclosure

l = the length of the enclosure.

Area of the enclosure = lw

(c)Width, w that would maximize the area = 162.5 feet

(d)Maximum Area

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240

step-by-step explanation:

graph using a few selected

x         y

-1       15

0         6

1         2.4

2       0.96