Question

find the greatest possible enclosed area of a rectangular corral given 400 feet of fencing

Answer 1

A square, a special rectangular form, requires the least fencing length.

Answer 2

So pretty much always with these problems:
1. Two equations
2. Solve for x or y in one of them.
3. Substitute into second equation and multiply out
4. Take the derivative and set to 0 to find critical points
5. test critical points on the appropriate original equation.

Thats pretty much how all of these go. So you want two equations first. The question wants area, so we definitely need that. We'll say

xy = area

Its mentionining enclosing, so it means perimeter. And since we have 400 feet of fenching, the perimeter will equal 400, meaning we can put this as our 2nd equation:

2x + 2y = 400.

Now we solve for x or y. Divide by 2
x + y = 200
y = 200-x

Next we substititue this into the other equation:

x(200-x) = -x^2+200x. Now if I take the derivative and set it to 0 I get

-2x + 200 = 0
x = 100

Because this is the only critical point, we dont have to worry about testing values. So now all we need to do is get y by pluggign this x into the perimeter equation we started with:

2(100) + 2y = 400
2y = 200
y = 100. So both x and y are 100, meaning the maximum area is (100)^2 = 10000ft

Answer 3

thank you!! @Psymon

Answer 4

Yep, np :3