Problem 28 - Essential Calculus

Question

A rectangular page is to contain 36 square inches of print. The margins on each side are to be \(1 \frac{1}{2}\) inches. Find the dimensions of the page such that the least amount of paper is used.

Step-by-Step Solution

Answer

The desired dimensions of the page are 9 inches by 9 inches.

Step 1: Define Variables

Let's first denote the dimensions of the printed part as \( x \) (width) and \( y \) (height). The dimensions of the page itself (including the margins) become \( x+3 \) (width) and \( y+3 \) (height), as we add \(1 \frac{1}{2}\) inches on each side.

Step 2: Setup the Equations

We are given that the print area is constant 36 square inches, which gives us the equation \( x*y = 36 \). Moreover, the total area of the page, which we aim to minimize, can be expressed as \( A =( x+3)*(y+3) \).

Step 3: Express the Area Equation in One Variable

We can express y in terms of x using the equation from Step 2: \( y = \frac{36}{x} \), and then substitute this into the area equation, resulting in \( A = x*\frac{36}{x} + 3*\frac{36}{x} + 3*x + 9=36+ \frac{108}{x} + 3x + 9 \)

Step 4: Find the Derivative of A

We differentiate the above equation with respect to x: \( A' = - \frac{108}{x^2} + 3 \).

Step 5: Set Derivative Equal to Zero to Find Minimum

We find the minimum by setting the derivative equal to zero: \( A' = - \frac{108}{x^2} + 3 = 0 \). Solving for x, we get \( x^2 = \frac{108}{3} \), and thus \( x = \sqrt{36} = 6 \).

Step 6: Calculate Corresponding y

Substitute \( x = 6 \) into the equation \( y = \frac{36}{x} \) we get \( y = 6 \).

Step 7: Calculate Dimensions of Page

Now, using \( x = 6 \) and \( y = 6 \), the dimensions of the page are \( x+3 = 9 \) inches width and \( y+3 = 9 \) inches height.