Question

An art page for a book is formatted like this: The artwork has an area of 30 square inches. The margins are 1 inch. Find x and y that minimize the total area of the paper.

Answer 1

To minimize the total area of the paper, the dimensions of the artwork should be x = sqrt(30) and y = sqrt(30).

Step-by-step explanation:

To minimize the total area of the paper, we need to find the dimensions of the artwork that minimize the combined area of the artwork and the margins.

Let x be the length of the artwork, and y be the width of the artwork.

The area of the artwork is given as 30 square inches, so we have xy = 30.

The total area of the paper is (x + 2)(y + 2), where 2 represents the 1-inch margins on each side.

To minimize the total area, we can take the derivative of the area function with respect to either x or y, set it equal to zero, and solve for one of the variables.

Let's take the derivative with respect to x:

dA/dx = (y + 2) - (x + 2) = 0

Simplifying this equation, we get y + 2 = x + 2, which implies that y = x.

Substituting y = x into the equation xy = 30, we get x^2 = 30, which gives us x = sqrt(30).

Therefore, the dimensions that minimize the total area of the paper are x = sqrt(30) and y = sqrt(30).