Solve the problem.
-The perimeter of a rectangle is 44 feet. Describe the possible lengths of a side if the area of the rectangle is to be greater than 105 square feet.
A) The length of the rectangle must lie between 7 and $15 \mathrm { ft }$
B) The length of the rectangle must be greater than $15 \mathrm { ft }$
C) The length of the rectangle must be greater than $15 \mathrm { ft }$ or less than $7 \mathrm { ft }$
D) The length of the rectangle must lie between 1 and $105 \mathrm { ft }$

Question

Quizplus · Accepted Answer

Given the perimeter of the rectangle is 44 feet, we can express the perimeter as $2l + 2w = 44$, where $l$ is the length and $w$ is the width. Simplifying, we get $l + w = 22$, or $w = 22 - l$. The area $A$ of the rectangle is given by $A = lw$, and we want this area to be greater than 105 square feet. Substituting $w$ from the perimeter equation, we get $A = l(22 - l) > 105$. This simplifies to a quadratic inequality $-l^2 + 22l - 105 > 0$. Solving this inequality for $l$ gives the range of lengths that satisfy the condition. The roots of the corresponding quadratic equation $-l^2 + 22l - 105 = 0$ are approximately 7 and 15, meaning the length $l$ must lie between these two values for the area to be greater than 105 square feet. Hence, the correct choice is that the length of the rectangle must lie between 7 and 15 feet.

Solve the Problem 15ft15 \mathrm { ft }15ft B) the Length of the Rectangle Must Be Greater Than

Solve the Problem $15 \mathrm { ft }$ B) the Length of the Rectangle Must Be Greater Than