A rectangular page is to contain 27 square inches of print. The page has to have a 3-inch margin on top and at the bottom and a 4-inch margin on each side (see figure). Find the dimensions of the page that minimize the amount of paper used.
The dimensions that minimize the amount of paper used are in.
(Simplify your answers. Use a comma to separate answers.)

Step 1 ：Let the width of the printed area be \(x\) inches and the height be \(y\) inches. Then the area of the printed area is \(xy = 27\) square inches.

Step 2 ：Since there is a 3-inch margin on the top and bottom, the total height of the page is \(y + 2 \times 3 = y + 6\) inches.

Step 3 ：Similarly, since there is a 4-inch margin on each side, the total width of the page is \(x + 2 \times 4 = x + 8\) inches.

Step 4 ：So, the total area of the page is \((x + 8)(y + 6)\) square inches.

Step 5 ：We want to minimize this total area. To do this, we can express \(y\) in terms of \(x\) using the equation \(xy = 27\), which gives us \(y = \frac{27}{x}\).

Step 6 ：Substituting \(y = \frac{27}{x}\) into the equation for the total area gives us \((x + 8)\left(\frac{27}{x} + 6\right)\).

Step 7 ：We can simplify this to \(27 + 6x + \frac{216}{x} + 48\), which simplifies further to \(6x + \frac{216}{x} + 75\).

Step 8 ：This is a function of \(x\) that we want to minimize. To find the minimum of a function, we can take its derivative and set it equal to zero.

Step 9 ：The derivative of \(6x + \frac{216}{x} + 75\) with respect to \(x\) is \(6 - \frac{216}{x^2}\). Setting this equal to zero gives us \(6 - \frac{216}{x^2} = 0\).

Step 10 ：Solving this equation for \(x\) gives us \(x = \sqrt{36} = 6\).

Step 11 ：Substituting \(x = 6\) into the equation \(y = \frac{27}{x}\) gives us \(y = \frac{27}{6} = 4.5\).

Step 12 ：So, the dimensions that minimize the amount of paper used are \(x + 8 = 6 + 8 = 14\) inches for the width and \(y + 6 = 4.5 + 6 = 10.5\) inches for the height.

Step 13 ：\(\boxed{14, 10.5}\) are the dimensions that minimize the amount of paper used.