A rectangular page is to contain 8 square inches of print. The margins at the top and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide. Find the dimensions of the page that will minimize the amount of paper used. (Let x represent the width of the page and let y represent the height.)

To find the dimensions of the page that will minimize the amount of paper used, we need to express the total amount of paper used as a function of the dimensions of the page.

Let's break down the problem step by step:

1. First, let's determine the area of the printed material. It is given that there should be 8 square inches of print. Since the printed material is in the shape of a rectangle, the area of the printed material can be expressed as the product of the width and height, which is 8 = x * y.

2. Next, let's consider the margins. It is stated that the margins at the top and bottom of the page are 2 inches wide, and the margins on each side are 1 inch wide. Therefore, the actual printable area of the page is reduced by 4 inches in width (2 inches on each side) and 4 inches in height (2 inches at the top and bottom). This means the width and height of the printable area would be (x - 4) and (y - 4), respectively.

3. To find the total amount of paper used, we need to find the total area of the page, which includes both the printed material and the margins. The width of the page will be the width of the printable area plus the total width of the margins, i.e., x = (x - 4) + 1 + 1, which simplifies to x = x - 2. Similarly, the height of the page will be the height of the printable area plus the total height of the margins, i.e., y = (y - 4) + 2 + 2, which simplifies to y = y - 2.

4. Now, let's substitute the value y = y - 2 derived from step 3 into our equation from step 1: 8 = x * (y - 2). Simplifying this equation will give us x in terms of y: x = 8 / (y - 2).

5. We now have x expressed in terms of y, so we can substitute this value into our equation x = x - 2 from step 3. Now, x = (8 / (y - 2)) - 2.

6. Finally, we have an equation for x in terms of y that represents the width of the page. To minimize the amount of paper used, we need to minimize the total area of the page. We can express the total area as the product of the width (x) and height (y) of the page: A = x * y.

7. Substitute the expression for x from step 5 into the total area equation from step 6: A = ((8 / (y - 2)) - 2) * y.

Now, we have the equation A = ((8 / (y - 2)) - 2) * y, which represents the total amount of paper used as a function of the height of the page (y). To find the dimensions of the page that will minimize the amount of paper used, we need to find the value of y that minimizes this equation. We can do this by taking the derivative of A with respect to y, setting it equal to zero, and solving for y.

Once we have the value of y, we can substitute it back into the equation for x from step 5 to find the corresponding value of x. This will give us the dimensions of the page that minimize the amount of paper used.

To minimize the amount of paper used, we need to minimize the total area of the page.

Let's start by calculating the actual printable area on the page. Since the top and bottom margins are 2 inches each and the side margins are 1 inch each, the printable width will be x - 2 - 2 = x - 4 inches and the printable height will be y - 1 - 1 = y - 2 inches.

To find the area of the printed portion, we multiply the width by the height:

Printed Area = (x - 4)(y - 2)

Since we know that the printed area should be 8 square inches, we can set up the following equation:

(x - 4)(y - 2) = 8

Next, we need to express one variable in terms of the other so that we can minimize the equation with one variable.

Let's solve the equation for y:

(x - 4)(y - 2) = 8

Distribute the x:
yx - 2x - 4y + 8 = 8

Rearrange the terms:
yx - 4y = 2x

Factor out y:
y(x - 4) = 2x

Divide both sides by (x - 4):
y = 2x / (x - 4)

Now we have y in terms of x.

To minimize the amount of paper used, we need to minimize the total area of the page, which is given by:

Total Area = (x)(y) = x(2x / (x - 4))

Now, we can find the derivative of the Total Area with respect to x and set it equal to zero to find the critical points:

d(Total Area)/dx = 0

Let's differentiate Total Area with respect to x:

d(Total Area)/dx = x(d(2x / (x - 4))/dx) + (2x / (x - 4))(d(x)/dx)

Simplifying the derivative:

2(x - 4) - 2x = 0

Simplifying further:

2x - 8 - 2x = 0

-8 = 0 - this equation is not possible.

Therefore, there are no critical points and we need to consider the endpoints of the feasible region.

Since we are dealing with dimensions, both x and y should be positive, which means x > 4 and y > 2.

Now, we can evaluate the Total Area at the endpoints of the feasible region and compare them to find the minimum area.

Case 1: x = 5
y = 2x / (x - 4) = 2(5) / (5 - 4) = 10 / 1 = 10

Total Area = x * y = 5 * 10 = 50 square inches

Case 2: x = 8
y = 2x / (x - 4) = 2(8) / (8 - 4) = 16 / 4 = 4

Total Area = x * y = 8 * 4 = 32 square inches

Case 3: x approaches positive infinity
As x approaches positive infinity, y approaches 2x / (x - 4) approaches 2, and the Total Area approaches infinity.

Comparing the areas:
The minimum area is obtained when x = 8 and y = 4, with a Total Area of 32 square inches.

Therefore, the dimensions of the page that minimize the amount of paper used are: Width = 8 inches, Height = 4 inches.

So, let the width of print be x, and the height be y.

xy = 8
y = 8/x

That means that the area of the page is

a = (x+2)(y+4) = (x+2)(8/x+4) = 4x+16+16/x

da/dx = 4-16/x^2

So, find where da/dx=0 for minimum area.

Note that I used x and y for the dimensions of the printed part. It just means that the equation will be different, but the answer will be the same.