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