Question

Shenelle has 100100100 meters of fencing to build a rectangular garden.

The garden's area (in square meters) as a function of the garden's width www (in meters) is modeled by:

A(w)=-(w-25)^2+625A(w)=−(w−25)

2

+625A, left parenthesis, w, right parenthesis, equals, minus, left parenthesis, w, minus, 25, right parenthesis, squared, plus, 625

What side width will produce the maximum garden area?

Answer 1

625

Explanation:

got it right on khan academy

proof:

Answer 2

The maximum area is 625 square meters.

Explanation:

We know that the area is determined by

`A(w)=-(w-25)^(2) +625`

To find the maxium area, we need to calculate the derivative of this function

`A'(w)=-2(w-25)`

Then, we make it equal to zero, to find a maxium value

`-2(w-25)=0`

Now, we solve for
`w`

`-2(w-25)=0\\w-25=0\\w=25`

But, according to the problem, the perimeter is 100 meters, because the fencing represents a perimeter.

`P=2(w+l)=100\\`

And,
`w=25`

So,

`2(25+l)=100\\25+l=50\\l=50-25\\l=25`

So, the maxium width is 25 meters, the maxium length is 25 meters, and the maxium area is the product of these dimensions

`A_(max) =25 * 25 =625 \ m^(2)`

Therefore, the maximum area is 625 square meters.