A cardboard poster containing 32 square inches of printed region is to have a margin of 2 inches at the top and bottom and 0.75 inch at the sides. The dimensions of the smallest piece of cardboard that can be used to make the poster are

A) $\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 3 }$ in by $\frac { 3 + 4 \sqrt { 3 } } { 2 }$ in
B) $\frac { 5 ( 3 + 4 \sqrt { 3 } ) } { 3 }$ in by $\frac { 3 + 4 \sqrt { 3 } } { 2 }$ ?in
C) $\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 3 }$ in by $\frac { 3 + 4 \sqrt { 3 } } { 3 }$ in
D) $\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 2 }$ in by $\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 2 }$ in
E) $\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 3 }$ in by $\frac { 3 + 4 \sqrt { 3 } } { 4 }$in

Question

A cardboard poster containing 32 square inches of printed region is to have a margin of 2 inches at the top and bottom and 0.75 inch at the sides. The dimensions of the smallest piece of cardboard that can be used to make the poster are

A) $\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 3 }$ in by $\frac { 3 + 4 \sqrt { 3 } } { 2 }$ in
B) $\frac { 5 ( 3 + 4 \sqrt { 3 } ) } { 3 }$  in by $\frac { 3 + 4 \sqrt { 3 } } { 2 }$ ?in
C) $\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 3 }$ in by $\frac { 3 + 4 \sqrt { 3 } } { 3 }$ in
D) $\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 2 }$  in by $\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 2 }$ in
E) $\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 3 }$  in by $\frac { 3 + 4 \sqrt { 3 } } { 4 }$in

Quizplus · Accepted Answer

The printed region is 32 square inches, and the margins add 2 inches to the height and 1.5 inches (0.75 inch on each side) to the width. Let the dimensions of the printed region be $x$ and $y$, so $xy = 32$. The dimensions of the cardboard are $x + 1.5$ and $y + 4$. To minimize the area of the cardboard, we set the derivative of the area with respect to one of the dimensions to zero, leading to the dimensions provided in choice A, which account for the optimal aspect ratio of the printed region plus the margins.

A Cardboard Poster Containing 32 Square Inches of Printed Region 4(3+43)3\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 3 }34(3+43​)​

A Cardboard Poster Containing 32 Square Inches of Printed Region $\frac { 4 ( 3 + 4 \sqrt { 3 } ) } { 3 }$