Find the dimensions of a rectangle of area 400 that has the smallest possible perimeter.
A)4 ft by 100 ft
B)1 ft by 400 ft
C)20 ft by 20 ft
D)2 ft by 200 ft

Question

Find the dimensions of a rectangle of area 400  that has the smallest possible perimeter.
A)4 ft by 100 ft 
B)1 ft by 400 ft 
C)20 ft by 20 ft 
D)2 ft by 200 ft

Quizplus · Accepted Answer

Let the length of the rectangle be $l$ and the width be $w$. We know that $lw = 400$. The perimeter of the rectangle is $2l + 2w$. Using the fact that $lw=400$, we can rewrite the perimeter as $2l + \frac{800}{l}$. Taking the derivative of this expression with respect to $l$ and setting it equal to $0$, we get:

$$2 - \frac{800}{l^2} = 0$$

Solving for $l$, we get $l=20$. Since the area is 400, the width must also be 20. Therefore, the dimensions of the rectangle are 20 ft by 20 ft. This gives us the smallest possible perimeter of 40 ft, making choice C the best choice.

Find the Dimensions of a Rectangle of Area 400