A rectangular playing field with a perimeter of 98 meters is to have an area of at least 444 square meters.Within what bounds must the length of the rectangle lie?

A) $444 m \leq L \leq 13 m$
B) $37 m \leq L \leq 98 m$
C) $12 m \leq L \leq 37 m$
D) $39 m \leq L \leq 14 m$
E) $444 m

Question

A rectangular playing field with a perimeter of 98 meters is to have an area of at least 444 square meters.Within what bounds must the length of the rectangle lie?
 
A) $444 m \leq L \leq 13 m$
B) $37 m \leq L \leq 98 m$
C) $12 m \leq L \leq 37 m$
D) $39 m \leq L \leq 14 m$
E) $444 m < L < 13 m$

Quizplus · Accepted Answer

Given a rectangle with a perimeter of 98 meters, we can express the perimeter as $2L + 2W = 98$, where $L$ is the length and $W$ is the width. Simplifying, we get $L + W = 49$. The area of the rectangle is given as at least 444 square meters, so $L \times W \geq 444$. To find the bounds for $L$, we can express $W$ in terms of $L$ from the perimeter equation: $W = 49 - L$. Substituting this into the area inequality gives $L(49 - L) \geq 444$, which simplifies to $49L - L^2 \geq 444$, and further to $L^2 - 49L + 444 \leq 0$.Solving the quadratic equation $L^2 - 49L + 444 = 0$ gives the roots as the possible bounds for $L$. The roots of this equation are $L = 12$ and $L = 37$, meaning the length of the rectangle must lie between these two values to satisfy both the perimeter and area requirements. Therefore, the length must lie within the bounds of 12 meters and 37 meters.

A Rectangular Playing Field with a Perimeter of 98 Meters 444m≤L≤13m444 m \leq L \leq 13 m444m≤L≤13m

A Rectangular Playing Field with a Perimeter of 98 Meters $444 m \leq L \leq 13 m$