Question

Find the greatest area of a rectangular garden that can be enclosed with 80 feet of fencing.
please go step by step

Answer 1

The closer to a square a given perimeter is, the more area there is. So, you would make a square for this garden, and a square has all equal sides. So, to find a side of the square, you divide 80 ft by 4, which gives you 20 ft. Multiply 20 ft by 20 ft to get your area.

Answer 2

I think you should explain the problem using calculus, since that is when you do these sorts of problems. I myself always had issues with these problems, so I'll let you handle it ^^ If you really can't though, I'll jump in to try and help

@jchiang

Answer 3

@smokeydabear can you help me please?

Answer 4

Okay okay I will try, give me a sec to remember.

Answer 5

Okay so to make this explanation short I'll assume you know what a derivative is. The answer would involve a square, but you may not always have an ideal situation (ie. weird triangles). Since we know we're dealing with a rectangle, we know right off the bat that the area of a rectangle is

L x W = A

We need to find the best combination of L and W that will give us the highest A. We can write the equation for the perimeter as

2L+2W=80

But to make this work out right, we want to eliminate one of the variables by substitution

2L=80-2W
L=40-W

Now that we have L in terms of W, we can rewrite the area formula as

W(40-W)=A
W^2+40W=A

You may not know it yet, but this is the ideal form we want to take the derivative of, since we only want one variable at a time. We need to find when the derivative is equal to 0, because if you can visualize a simple graph of the area, the slope will be 0 at its highest point (similar to how a ball thrown into the air will reach its max height and at the same time have a velocity of 0, because it has just started to fall back down). So the derivative, which will be set to 0, is

2W+40=0
2W=40
W=20

We just found what the W has to be to get the highest possible area out of this rectangle. Since this is a rectangle, that means two sides will be 20ft (top and bottom make up the width). We can just use common sense to figure out what the other two sides have to be, since they must be equal to each other, but let's stick to the normal method. Since W=20, we can plug that back into the equation we had for L way back up

L=40-(20)
L=20

And obviously that means both sides are 20ft. As you can see, since L=W, we have a square, as jchiang said. Hope that helps! If there's anything you need me to clarify, I'm here!