Answer
Possible sides (in feet) are in $(0, 10]$ or $[80, 90)$
Work Step by Step
Let $P$ be the perimeter of a rectangle.
As we are given $P = 180$ feet.
The perimeter $P$ of a rectangle is given as follows:
$P = 2L + 2W$
This implies
$180 = 2L + 2W$
$180 = L + W$
$L = 90 – W$ … (i)
Let A be the area of a rectangle.
As we are given $A ≤ 800$ square feet
The area $A$ of a rectangle is given as follows:
$A = L × W$
As per the question, the area, $A = L × W$ must not be exceeded by $800$ ft.
Then
$L × W ≤ 800$
Substitute $L = 90 – W$ from equation (i).
$(90-W) × W ≤ 800$
$90W - W^{2} ≤ 800$
$W^{2} – 90W + 800 ≤ 0$
$W2 – 80W - 10W + 800 ≤ 0$
$W (W – 80) -10(W – 80) ≤ 0$
$(W- 80) (W – 10) ≤ 0$
$W = 80$ ft or $10$ ft
Now, calculate $L$.
$L = 90 – W$
This implies
$L = 90 – 80 = 10$ ft
or
$L = 90 – 10 = 80$ ft
Hence, $L = 80$ ft or $10$ ft and $W = 10$ ft or $80$ ft.
Since $A ≤ 800$ sq ft, any length $L ≤ 80$ ft will be the possible length of a side.
However, the inequality involves (less than or equal to), we must also include the solution of $W^{2} – 90W + 800 ≤ 0$, namely $80$, $10$, in the solution set.
Therefore, we conclude that the possible sides (in feet) are in $(0, 10]$ or $[80, 90)$.
