Question

A farmer has 1000 feet of fence to enclose a rectangular area. what dimensions for the rectangle result in the maximum area enclosed by the fence

Answer 1

Very much like the problem we just did.
Assume the length of the rectangle to be L and the Width to be W.
What is its perimeter in terms of L and W?

Answer 2

(2L)(2W)

Answer 3

sorry plus

Answer 4

(2L)+(2W)

Answer 5

Yes. Equate that to 1000 and find W in terms of L.

Answer 6

(L-1000)= W?

Answer 7

Perimeter = 2L + 2W = 2(L + W) = 1000
W = ?

Answer 8

2L-1000=W?

Answer 9

First divide both sides of the equation by 2.
Then solve for W.

Answer 10

i have L+W=500 but i don't know how to get rid of the L but i have a feeling that i should divide the 500 by 2 and ill get 250

Answer 11

L + W = 500
We are not solving for L. We just want to find W in terms of L
Subtract L from both sides of the equation and you will have W = ?

Answer 12

so i would have 4W=250

Answer 13

sorry 4w=100 but then i would have to divide by four on both sides

Answer 14

No.
L + W = 500
Subtract L from both sides.
W = 500 - L

Answer 15

oh okayi see

Answer 16

So we have a rectangle whose
Length = L
Width = (500 - L)

What is the area A of the rectangle?

Answer 17

(L)(W)

Answer 18

Yes. Area = LW = L(500 - L)

Follow?

Answer 19

would i have to distribute?

Answer 20

Not necessary.
We need to maximize the area which means we have to maximize the product
L(500 - L)

But you will notice the similarity to the previous problem where the product of two numbers x(30 -x) has to be maximized and the answer was x = 30/2 = 15.

In the same way the product L(500 - L) will be maximum when L = 500/2 = 250.

Follow?

Answer 21

some what

Answer 22

Previously we plugged in various values of x from 1 to 17 and determined the maximum. But with a large number such as 500 it is going to be too time consuming without using calculus. So whenever a perimeter of a rectangle is fixed (as in this problem when perimeter = 1000) you will get maximum area when the length and the width of the rectangle are the same. In other words the area has to be a square. So if you put L = 250 in the equation L+W = 500 we had earlier you will find W is also 250. So Area is maximum when L = W = 250 Area = LW = (250)(250) = 62,500.

Answer 23

oh okay well ill b

Answer 24

racticing with oth

Answer 25

Alright.

Answer 26

thank you

Answer 27

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