How do you do this?

A farmer has 336 feet of fencing and wants to build two identical pens for his prize-winning pigs. The pens will be arranged as shown. Determine the dimensions of a pen that will maximize its area.

"as shown" doesn't work here.

Are there two pens with a common side?

Yes

Ok, in your diagram,

let the length of the entire enclosure be y ft
let the width be x ft (there are 3 of these)

3x + 2y = 336
y = (336 - 3x)/2 = 168 - 3x/2

area = xy
= x(168 - 3x/2)
= 168x - (3/2)x^2

this is a downwards opening parabola
the x of the vertex is -168/-3 = 56
then y = 168 - (3/2)(56) = 84

the length is 84 ft, and each of the 3 width = 56 ft

check: 2(84) + 3(56) = 636

Well, first of all, let me just say that I'm glad the farmer is taking pig accommodations seriously. Now, let's get down to business.

Since we're dealing with two identical pens, we can divide the total amount of fencing by 2 to find the amount of fencing for each pen. So each pen will have 168 feet of fencing *cue dramatic music*.

To maximize the area of the pen, we need to find the dimensions that make that happen. Let's break it down, shall we?

Let's label one side of the pen as "x" and the adjacent side as "y". Since the opposite sides of each pen are going to share fencing, we only need to focus on three sides. So, for each pen, we have 2x (each pen has two sides of length x) and y (the shared side).

Now, we know the perimeter of each pen is 168 feet, so we can set up the equation:
2x + y + y = 168.

Simplifying this equation, we get:
2x + 2y = 168,
x + y = 84.

To maximize the area, we need to solve for y in terms of x. So we get:
y = 84 - x.

Now, we can calculate the area of each pen by multiplying the length by the width. The area is:
A = x * (84 - x).

To find the dimensions that will maximize the area, we need to find the maximum value of this quadratic equation. You can do that by using calculus (which is not my strong suit), or you can use a graphing calculator to find the vertex of the parabola.

And there you have it! The dimensions of the pen that will maximize its area. Remember, happy pigs make for happy farmers!

To determine the dimensions of a pen that will maximize its area, we can use calculus and the concept of optimization. Here's how you can solve this problem step by step:

1. Let's assume the outer length of each pen is x feet. Since there are two identical pens arranged in the given way, the total length of fencing used for the outer perimeter of the pens will be 2x.

2. The total length of the fencing used for the inner division between the pens is also 2x.

3. Therefore, the total length of fencing used would be 2x + 2x = 4x.

4. According to the problem, the farmer has a total of 336 feet of fencing available, so we can set up the equation: 4x = 336.

5. Solve the equation to find the value of x: divide both sides by 4, x = 84.

6. Now, we have the length of the outer side of each pen, which is 84 feet. To find the width, we can divide the remaining fencing by 2, since it is shared by both pens: (336 - 2x) / 2 = (336 - 2*84) / 2 = 84 / 2 = 42.

7. Therefore, the dimensions of each pen that will maximize its area are 84 feet by 42 feet.

By constructing the pens with these dimensions, the farmer will utilize all 336 feet of fencing and achieve the maximum area possible.

hey y'all

the length is 45
width is 30
length is 40 cause its on the brainy duh but since theres 4 adjacent sides that'd be 45*4 which is 180 then u subtract 180 from 360 and you get 180 so you divide that by six for the non adjacent sides