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?
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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 the perimeter $P = 2l + 2w = 98$ meters and the area $A = lw \geq 444$ square meters, we can solve for one variable in terms of the other using the perimeter equation and then substitute into the area inequality. Let's solve for $w$ in terms of $l$ from the perimeter equation: $w = \frac{98 - 2l}{2} = 49 - l$. Substituting this into the area inequality gives $l(49 - l) \geq 444$, which simplifies to $l^2 - 49l + 444 \leq 0$. Solving this quadratic inequality for $l$ gives the bounds for $l$. The roots of the corresponding equation $l^2 - 49l + 444 = 0$ can be found using the quadratic formula, yielding $l = 12$ and $l = 37$ as the critical points. Therefore, the length $l$ must lie between 12 meters and 37 meters, inclusive, to satisfy both the perimeter and area requirements.

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 $444 m \leq L \leq 13 m$