Question

A rectangular playground is to be fenced off and divided into two parts by a fence parallel to one side of the playground. 840 feet of fencing is used. Find the dimensions of the playground that will enclose the greatest total area.

Answer 1

Answer:

The dimensions of the playground that will enclose the greatest total area are 140 ft x 210 ft.

Step-by-step explanation:

We have a rectangular with sides "a" and "b", so that the area is:

$S=a\cdot b$

The perimiter for this rectangle is

$P=2(a+b)$

The fence is for the perimeter plus the division, which has a length of "a".

So the total fencing is:

$F=P+a=(2a+2b)+a=3a+2b=840$

We can express one side in function of the other, in order to optimize the area.

$3a+2b=840\\\\2b=840-3a\\\\b=420-(3/2)a$

Then, we can write the area as:

$S=a\cdot b=a*(420-(3/2)a)=-(3/2)a^2+420a$

To maximize the area we will derive and equal to zero

$dS/da=-3a+420=0\\\\3a=420\\\\a=420/3=140$

Then, the value for the other side of the rectangle is:

$b=420-(3/2)*140=420-210=210$