Question

7. A rectangular page is to contain 30 square inches of print. The margins on each side (all 4 sides of the printed part) are 1 inch. Find the dimensions of the page such that the least amount of paper is used.

Answer 1

To find the dimensions of the page that will use the least amount of paper, we need to minimize the area of the page while still accommodating the 30 square inches of print and the 1-inch margins on all sides. The dimensions of the page that will use the least amount of paper are approximately √30 inches by √30 inches.

Step-by-step explanation:

To find the dimensions of the page that will use the least amount of paper, we need to minimize the area of the page while still accommodating the 30 square inches of print and the 1-inch margins on all sides. Let's assume the width of the printed part is x inches. Since the margins on all sides are 1 inch, the width of the page will be x + 2 inches. Similarly, the length of the page will be y + 2 inches.

The area of the page is given by (x + 2)(y + 2) square inches. We want to minimize this area. We are given that the printed part contains 30 square inches of print, so the area of the printed part is xy square inches.

Therefore, the total area of the page, including the margins, is given by (x + 2)(y + 2) = xy + 2x + 2y + 4 square inches. To minimize this area, we can take its derivative with respect to x and set it equal to 0.

dA/dx = y + 2 = 0

Therefore, y = -2. Since we can't have a negative length for the page, we ignore this solution. Thus, the dimensions of the page that will use the least amount of paper are x + 2 by y + 2, or x + 2 by x, which simplifies to x by x.

From the earlier calculation, we know that xy = 30. Therefore, x^2 = 30, and taking the square root of both sides, we get x = √30 inches.

The dimensions of the page that will use the least amount of paper are approximately √30 inches by √30 inches.

Answer 2

The dimensions of the page that use the least amount of paper are approximately 4.15 inches by 4.15 inches.

Step-by-step explanation:

To find the dimensions of the page that will use the least amount of paper, we need to minimize the area of the page. Let's assume the width of the printed part of the page is 'x' inches. Given that there is a 1-inch margin on each side, the total width of the page would be 'x + 2' inches.

Similarly, let's assume the height of the printed part of the page is 'y' inches. The total height of the page would then be 'y + 2' inches.

Since the dimensions of the printed part and the total page are known, we can set up the following equation to find the area:

(x + 2)(y + 2) = 30

Expanding the equation, we get:

xy + 2x + 2y + 4 = 30

Combining like terms, we have:

xy + 2x + 2y = 26

We can rewrite the equation as:

x(y + 2) + 2(y + 2) = 26

Factoring out the common terms, we get:

(y + 2)(x + 2) = 26

Simplifying further:

y + 2 = 26 / (x + 2)

y = (26 / (x + 2)) - 2

To minimize the amount of paper used, we need to find the dimensions that minimize the values of 'x' and 'y'. We can do this by graphing the equation or using calculus techniques such as finding critical points and doing optimization.

For example, if we graph the equation, we will see that the dimensions that minimize the area of the page are approximately x = 2.15 inches and y = 2.15 inches. Therefore, the dimensions of the page that use the least amount of paper are approximately 4.15 inches by 4.15 inches.