Q. 54 - Calculus

Question

Alina needs to make a flyer for her band’s concert. The flyer must contain $20$ square inches of printed material and for design, purposes should have side margins of $1$inch and top and bottom margins of $2$ inches. What size paper should Alina use in order to use the least amount of paper per flyer possible?

Step-by-Step Solution

Answer

Ans:  The dimensions of the paper are $6.47 in. \times 8.47 in.$

Step 1. Given information.

given,

     The figure above shows a paper of width x inches and of height y inches.

The top and bottom margins are 2 inches each and are shown by horizontal dashed lines.

The side margins are 1 inch and are shown by the vertical dashed lines in the figure above.

Step 2. The objective is to find the dimensions of the paper such that the least amount of paper is used for 20 square inches of the printing area.

From the figure, above it can be seen that the width of the printable area is, $x-1-1=x-2$

The height of the printable area is, $y-2-2=y-4$

The printable area is, $(x-2)(y-4)$.

The area should be $20$ square inches. Therefore, $(x-2) (y-4)=20$.

The perimeter of the paper is,

    $S=2(x+y)$

Substitute for $y$ in terms of $x$,

      $\begin{array}{r}S\left(x\right)=2\left(x+\frac{20}{x-2}+4\right)\\ =2x+\frac{40}{x-2}+8\end{array}$

Step 3. Differentiate the function with respect to x .

   $\begin{array}{r}2-\frac{40}{(x-2{)}^2}=0\\ (x-2{)}^2=20\\ x\approx 6.47\end{array}$

The value of $y$ is,

   $y=\frac{20}{6.47-2}+4 \approx 8.47$

Therefore, the dimensions of the paper are $6.47 in. \times 8.47 in.$