A book designer decided that the pages of a book should have 1-in. margins at the top and bottom and

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in. margins on the sides. She further stipulated that each page should have an area of 50 in.2. Find a function in the variable x, giving the area of the printed region (see the figure).

To find a function in the variable x that gives the area of the printed region, we need to understand the dimensions and layout of the page.

Let's start by drawing a diagram to visualize the page:

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| Printed |
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The top and bottom margins are each 1 inch, so the height of the printed region will be reduced by 2 inches (1 inch from the top and 1 inch from the bottom). Therefore, the height of the printed region is (x - 2) inches.

The side margins are each 1/2 inch, so the width of the printed region will be reduced by 1 inch (1/2 inch from the left and 1/2 inch from the right). Therefore, the width of the printed region is (x - 1) inches.

The area of a rectangle is given by the formula A = length * width. In this case, the length is the height of the printed region (x - 2) inches and the width is the width of the printed region (x - 1) inches.

Therefore, the function that gives the area of the printed region as a function of x is:

A(x) = (x - 2) * (x - 1)

Simplifying this equation will give the desired function in the variable x representing the area of the printed region.

To find the function that gives the area of the printed region, we need to consider the total area of the page and subtract the area of the margins.

Let's assume the dimensions of the printed region are x inches by y inches.

The total dimensions of the page would be:
Length = x + 2(1/2) inches (2 margins of 1/2 inch each)
Width = y + 2(1) inches (2 margins of 1 inch each)

The area of the total page would be:
Area of page = Length * Width
Area of page = (x + 1) * (y + 2)

Since the stipulated area of the page is 50 in², we have the equation:
(x + 1) * (y + 2) = 50

Now, we can solve this equation for y in terms of x:

(x + 1) * (y + 2) = 50
(y + 2) = 50 / (x + 1)
y = 50 / (x + 1) - 2

The function that gives the area of the printed region in terms of x is:
Area(x) = x * y
Area(x) = x * (50 / (x + 1) - 2)

Therefore, the function in the variable x, giving the area of the printed region, is:
Area(x) = x * (50 / (x + 1) - 2)