A pigpen is to be enclosed using 640 m of fencing along three of its sides, the fourth side being a barn. Determine the dimensions of the pen that will maximize the area (100 PTS + BRAINLIEST)

Since we know we only have 640 feet of fence available, we know that L + W + L = 640, [tex]\implies [/tex] 2L + W = 640. This allows us to represent the width, W, in terms of L: W = 640 – 2L

Remember, the area of a rectangle is equal to the product of its width and length, therefore,

[tex]A_{\rm pigpen}=L(640-2L)\implies 640L-2L^2[/tex]

Notice that, quadratic has been vertically reflected, since the coefficient on the squared term is negative, so the graph will open downwards, and the vertex will be a maximum value for the area.

recall,

[tex]x_{max}=-b/2a[/tex]
[tex]f(x)_{max}=f(x_{max})[/tex]

Since our function is A(L)=640L-2L², we get

a=-2
b=640

plug in the value of a and b into the first formula:

[tex]L_{max}=-(640)/2(-2)\implies 160 [/tex]

[tex]A(L)_{max}=640(160)-2(160)^2\implies 51200 [/tex]

hence,the dimensions of the pen that will maximize the area are L=160m and W=51200/160=320m

and we're done!

semsee45 semsee45 · Answer 2 · 2022-04-23T16:23:03+02:00

Answer:

width (x) = 160 m

length (y) = 320 m

(refer to the attached diagram)

Step-by-step explanation:

**see attached diagram**

Let x = width of the pig pen

Let y = length of the pig pen

(where length > width)

Given:

Fence = 640 m
4th side is the barn

⇒ 2x + y = 640 m

⇒ y = 640 - 2x

Area of the pig pen = width × length

⇒ f(x) = xy

Substituting y for y = 640 - 2x:

⇒ f(x) = x(640 - 2x)

= 640x - 2x²

Now we have a function for the area of the pig pen, to find the value of x that will maximize the area, differentiate the function:

⇒ f'(x) = 640 - 4x

Set it to zero and solve for x:

⇒ f'(x) = 0

⇒ 640 - 4x = 0

⇒ 640 = 4x

⇒ x = 160

Substitute the found value of x into y = 640 - 2x to find the length:

⇒ y = 640 - 2(160) = 320

Therefore, the dimensions of the pen that will maximize the area are:

width (x) = 160 m
length (y) = 320 m