Answer:
Option A is correct
The maximum area is, 6889 square feet
Step-by-step explanation:
Perimeter of a rectangle is given by:
\(P = 2(l+w)\)
where,
l is the length and w is the width of the rectangle respectively.
As per the statement:
You have 332 feet of fencing to enclose a rectangular region.
⇒\(P = 332\) feet
Then using perimeter formula:
\(332 =2(l+w)\)
Divide both side by 2 we get;
\(166 = l+w\)
or
\(l=166-w\) .....[1]
Area of a rectangle(A) is given by:
\(A = lw\) .....[2]
Substitute the value of [1] in [2] we have;
\(A = (166-w)w\)
⇒\(A = 166w -w^2\)
We have to find the maximum area:
A quadratic equation \(y=ax^2+bx+c\) then the axis of symmetry is given by:
\(x = -\frac{b}{2a}\)
The maximum area occurs at:
\(w = -\frac{166}{2(-1)} = \frac{166}{2} = 83\) feet
Substitute the value of w in [1] we have;
\(l = 166-83 = 83\) feet
Substitute the value of l and w in [2] we have;
\(A_{max} = 83 \cdot 83 = 6889\) square feet
therefore, the maximum area is, 6889 square feet
