Find the dimensions of a rectangle of area 64 $\mathrm{t}^{2}$ that has the smallest possible perimeter.
A) 4 ft by 16 ft
B) 2 ft by 32 ft
C) 1 ft by 64 ft
D) 8 ft by 8 ft

Question

Quizplus · Accepted Answer

Let the dimensions of the rectangle be $l$ and $w$, then we have $lw=64$. We want to minimize the perimeter $P=2l+2w$. Using AM-GM inequality, we have
$$P=2l+2w\geq 4\sqrt{lw}=4\sqrt{64}=32.$$
The equality holds if and only if $2l=2w$ and $lw=64$, which implies $l=w=8$. Therefore, the minimum perimeter is $32$ and it is achieved when the dimensions of the rectangle are $8$ by $8$. Thus the answer is choice (D).

Find the Dimensions of a Rectangle of Area 64 t2\mathrm{t}^{2}t2 That Has the Smallest Possible Perimeter

Find the Dimensions of a Rectangle of Area 64 $\mathrm{t}^{2}$ That Has the Smallest Possible Perimeter