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

Answer:

The maximum area is 625 square meters.

Step-by-step 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

$A_{max} =25 \times 25 =625 \ m^{2}$

Therefore, the maximum area is 625 square meters.

Answer 2

Answer:

625

Step-by-step explanation:

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proof: