a page is to contain 30 square inches of print. the margins at the top and bottom are each 2 inches wide. the margin on each side are only 1 inch wide. find the dimensions of the page so that the least paper is used

To find the dimensions of the page that use the least amount of paper, we can start by calculating the dimensions of the printable area on the page.

Given:
Margin at the top and bottom = 2 inches
Margin on each side = 1 inch
Total printable area = 30 square inches

Let's assume the width of the page is x and the height of the page is y.

The width of the printable area is x - 2 inches on each side, so it will be (x - (2 + 2)) inches = (x - 4) inches.

The height of the printable area is y - 2 inches at the top and bottom, so it will be (y - (2 + 2)) inches = (y - 4) inches.

The area of the printable area is given by the equation:
Area = width × height
30 square inches = (x - 4) inches × (y - 4) inches

Now we need to minimize the amount of paper used, which means we need to minimize the total area. We can do this by finding the dimensions that make the printable area as close to a square as possible.

Let's assume that the dimensions of the printable area are equal, so x - 4 = y - 4.
This implies x = y.

Substituting x = y in the equation, we have:
30 square inches = (x - 4) inches × (x - 4) inches
30 square inches = (x - 4)²

Taking the square root of both sides, we get:
√30 square inches = ±(x - 4) inches

Since the dimensions cannot be negative, we take the positive square root:
√30 square inches = x - 4 inches

Simplifying further, we have:
x = √30 + 4 inches

Therefore, the width and height of the page that use the least amount of paper are both √30 + 4 inches.

To find the dimensions of the page that use the least amount of paper while still containing 30 square inches of print, you can follow these steps:

1. Determine the width of the printable area:
- Subtract the total margin width on each side (2 inches on the top and bottom, and 1 inch on each side) from the total width of the page. Let's call this width W.

W = Total width of the page - Total margin width
W = Total width of the page - (2 inches + 2 inches + 1 inch + 1 inch)
W = Total width of the page - 6 inches

2. Determine the height of the printable area:
- Subtract the total margin height (2 inches on the top and bottom) from the total height of the page. Let's call this height H.

H = Total height of the page - Total margin height
H = Total height of the page - (2 inches + 2 inches)
H = Total height of the page - 4 inches

3. Use the given area of 30 square inches (length × width) to find a relationship between the width and height of the printable area:
- Multiply the length (W) by the width (H) to get 30 square inches.

WH = 30 square inches

4. Substitute the expressions for W and H from steps 1 and 2 into the equation WH = 30, obtained in step 3:

(Total width of the page - 6 inches) × (Total height of the page - 4 inches) = 30

Simplify and rearrange the equation to solve for the unknown total width and total height.

Total width of the page × Total height of the page - (10 inches × Total width of the page + 4 inches × Total height of the page) + 24 inches = 30

Total width of the page × Total height of the page - 10 inches × Total width of the page - 4 inches × Total height of the page + 24 inches = 30

Total width of the page × Total height of the page - 10 inches × Total width of the page - 4 inches × Total height of the page = 6 inches

(Total width of the page - 4 inches) × (Total height of the page - 10 inches) = 6 square inches

5. Since we're trying to minimize paper usage, we want to find the dimensions that result in the smallest page area. To minimize the area, we want the dimensions to be as close as possible. One way to achieve this is by making the difference between the total width and the total height as small as possible while still satisfying the equation. In this case, the closest values are 2 inches and 3 inches.

Total width of the page - 4 inches = 2 inches
Total height of the page - 10 inches = 3 inches

6. Solving for the total width and total height gives:

Total width of the page = 2 inches + 4 inches
Total width of the page = 6 inches

Total height of the page = 3 inches + 10 inches
Total height of the page = 13 inches

So, for the least paper usage while containing 30 square inches of print, the dimensions of the page should be 6 inches in width and 13 inches in height.

If the print is x by y, then the page is (x+2)(y+4)

Since xy=30, we want to minimize

a = (x+2)(30/x + 4) = 4x+8 + 60/x

So, find x when da/dx = 0 to get the minimum area.