Question

A rectangular playground is to be fenced off and divided into two parts by a fence parallel to one side of the playground. 480 feet of fencing is used. find the dimensions of the playground that will enclose the greatest total area

Answer 1

120 by 80 feet with the divider 80 feet long

Explanation:

Let’s consider a rectangular playground with length L and width W. The playground is divided into two parts by a fence parallel to one side of the playground. The total amount of fencing used is 2L + 3W = 480 feet.

To find the maximum value of the total area A = LW, we can express W in terms of L: W = (480 − 2L)/3. Substituting this expression for W into the expression for A, we get A = L(480 − 2L)/3 = (480L − 2L^2)/3.

To find the maximum value of A, we can take its derivative and set it equal to zero: dA/dL = (480 − 4L)/3 = 0. Solving for L, we get L = 120. Substituting this value of L into the expression for W, we get W = (480 − 2L)/3 = (480 − 240)/3 = 80.

Therefore, the dimensions of the playground that will enclose the greatest total area are 120 feet by 80 feet with the divider 80 feet long.

Answer 2

For the greatest total area, length will be 120 ft. and width will be 80 ft.

Explanation

Lets assume, the length of the playground is
`l` ft. and width is
`w` ft.

Suppose, the playground is divided into two parts by a fence parallel to width. That means the length of the divider fence will be
`w` ft.

So, the total length of the fence needed
`=(2l+3w) ft.`

It is given in the question that 480 feet of fencing is used. That means...

`2l+3w= 480\\ \\ 2l= 480-3w\\ \\ l=240-(3)/(2)w ...............................(1)`

Now, the area of the playground....

`A= l*w\\ \\ A=(240-(3)/(2)w)*w\\ \\ A= 240w-(3)/(2)w^2`

Taking derivative on both side in respect of
`w` , we will get...

`(dA)/(dw)=240-(3)/(2)(2w) \\ \\ (dA)/(dw)=240-3w`

A will be maximum when
`(dA)/(dw)=0` . That means...

`240-3w=0\\ \\ 3w=240\\ \\ w= (240)/(3)=80`

Now plugging this
`w=80` into equation (1)...

`l= 240-(3)/(2)(80) \\ \\ l=240-120=120`

So, for the greatest total area, length will be 120 ft. and width will be 80 ft.