5. A rectangular page is to contain 24 square inches of print.The margins at the top and bottom of the page are to be 1.5 inches and the margins on the right and left are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used?

## Step 1<br />The problem involves optimizing the dimensions of a rectangular page to minimize the amount of paper used. The area of the printed part is given as 24 square inches. The margins on the top and bottom are 1.5 inches each, and the margins on the right and left are 1 inch each. <br /><br />## Step 2<br />Let's denote the dimensions of the printed part as \(x\) and \(y\), where \(x\) is the width and \(y\) is the height. The total area of the page, including the margins, is then \((x + 2)(y + 3)\).<br /><br />## Step 3<br />The area of the printed part is \(xy = 24\). We can express \(y\) in terms of \(x\) as \(y = \frac{24}{x}\).<br /><br />## Step 4<br />Substituting \(y\) into the equation for the total area get \((x + 2)\left(\frac{24}{x} + 3\right)\).<br /><br />## Step 5<br />To minimize this expression, we take the derivative with respect to \(x\) and set it to zero. Solving for \(x\), we find \(x = 4\).<br /><br />## Step 6<br />Using this value, \(y = \frac{24}{4} = 6\).<br /><br />## Step 7<br />Therefore, the dimensions of the page that minimize the amount of paper used are \(x = 4\) inches and \(y = 6\) inches.