A rectangular page is to contain square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Question

A rectangular page is to contain  square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Quizplus · Accepted Answer

Let the length of the page be $x$ and the width be $y$. We know that the area of the printed region is  $11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11$ square inches. Since there are 1-inch margins on each side, the actual printed region has dimensions $(x-2)$ by $(y-2)$. So, we have the equation:

$$(x-2)(y-2)=11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11$$

Expanding this equation, we have:

$$xy-2x-2y+4=11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11$$

We want to minimize the amount of paper used, which is just the area of the page, given by $xy$. So, we want to minimize the expression $xy$ subject to the constraint:

$$xy-2x-2y+4=11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11$$

We can rearrange this constraint to get:

$$xy=2x+2y-11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11+4$$

Substituting $u=x-2$ and $v=y-2$, we get:

$$(u+2)(v+2)=11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11$$

Expanding and simplifying, we get:

$$uv+2u+2v+4=11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11$$

We can rearrange this equation to get:

$$uv=11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11-2u-2v-4$$

Now, we want to minimize the expression $uv$ subject to the constraint:

$$uv=11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11-2u-2v-4$$

We can use AM-GM inequality to get:

$$\frac{u+v}{2}\geq\sqrt{uv}$$

Rearranging, we get:

$$uv\leq\left(\frac{u+v}{2}ight)^2$$

Substituting $p=u+v$, we get:

$$uv\leq\left(\frac{p}{2}ight)^2$$

So, for a fixed value of $p$, the value $uv$ is minimized when $u=v$ (i.e. when the page is a square). Therefore, we want to find the factors of $11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11$ that are as close together as possible. We can see that $11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11$ is close to $120^2$, and indeed we have:

$$11ec1854_fdb8_5ca0_acfc_979e861e84c5_TBX8702_11=120^2-4$$

So, $u=v=118$, and the dimensions of the page are $x=u+2=120$ and $y=v+2=120$. Therefore, the answer is $\boxed{	extbf{(A)}}$.

A Rectangular Page Is to Contain Square Inches of Print