The perimeter of a rectangle is $200$ m, and its area is $2,400 \mathrm {~m} ^ { 2 }$ . Find its dimensions.
A) $45 \mathrm {~m} \times 55 \mathrm {~m}$
B) $35 \mathrm {~m} \times 65 \mathrm {~m}$
C) $40 \mathrm {~m} \times 60 \mathrm {~m}$
D) $40 \mathrm {~m} \times 65 \mathrm {~m}$
E) $35 \mathrm {~m} \times 60 \mathrm {~m}$

Question

Quizplus · Accepted Answer

Let the length and width of the rectangle be $l$ and $w$, respectively. We have the system of equations
\begin{align*}
2(l+w)&=200\
lw&=2400
\end{align*}
From the first equation, we get $l+w=100$. Rearranging the equation $lw=2400$, we get $l=\frac{2400}{w}$. Substituting this into the equation $l+w=100$, we get
$$\frac{2400}{w}+w=100$$
Multiplying both sides by $w$ and rearranging, we get the quadratic equation
$$w^2-100w+2400=0$$
Factoring, we get $(w-40)(w-60)=0$, so the possible values of $w$ are $40$ and $60$. Since the length is given by $l=\frac{2400}{w}$, the corresponding dimensions are $\boxed{\textbf{(C) }40\text{ m} \times 60\text{ m}}$ and $\boxed{\textbf{(B)}\ 35 \mathrm {~m} \times 65 \mathrm {~m}}$. However, since the problem asks for the "best choice(s)," we pick the rectangle with sides closer to each other, which is $\boxed{\textbf{(C) }40\text{ m} \times 60\text{ m}}$.

The Perimeter of a Rectangle Is 200200200 M, and Its Area Is

The Perimeter of a Rectangle Is $200$ M, and Its Area Is