Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 100 square meters.
A)Dimensions:
B)Dimensions:
C)Dimensions:
D)Dimensions:
E)Dimensions:

Question

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 100 square meters.
A)Dimensions: 
B)Dimensions: 
C)Dimensions: 
D)Dimensions: 
E)Dimensions:

Quizplus · Accepted Answer

Let the length of the rectangular solid be $l$, the width be $w$, and the height be $h$. Since the base is a square, we know that $l=w$. The surface area is given by $lw+lh+wh=100$. Since $l=w$, we have $2l^2+h(2l)=100\Rightarrow h=\frac{50-l^2}{l}$. Thus, the volume of the rectangular solid is $V=l^2h=l^2\cdot\frac{50-l^2}{l}=50l-l^3$. To find the maximum volume, we take the derivative of $V$ and set it equal to zero: $\frac{dV}{dl}=50-3l^2=0\Rightarrow l=\sqrt{\frac{50}{3}}\approx4.08$. Therefore, the dimensions of the rectangular solid are $4.08$ meters by $4.08$ meters by $\frac{50-(4.08)^2}{4.08}\approx7.85$ meters, which matches choice D.

Determine the Dimensions of a Rectangular Solid (With a Square