A book designer decided that the pages of a book should have 1-in. margins at the top and bottom and
1
2
in. margins on the sides. She further stipulated that each page should have an area of 52 in.2. Find a function in the variable x, giving the area of the printed region
To find a function in terms of the variable x that gives the area of the printed region, we can start by visualizing the dimensions of the printed region.
Let's assume that the length of the book is L and the width of the book is W. With 1-inch margins at the top and bottom, the height of the printed region would be (L - 2). Similarly, with 1/2-inch margins on the sides, the width of the printed region would be (W - 1).
The area of a rectangle is given by the formula A = length x width. In this case, the length is (L - 2) and the width is (W - 1). So, the equation for the area A of the printed region is:
A = (L - 2)(W - 1)
However, we have an additional condition that the area of the printed region should be 52 in². Therefore, we can set up the following equation:
(L - 2)(W - 1) = 52
Now, we have an equation with two variables (L and W), but we need to express the area A in terms of a single variable. We can do this by rearranging the equation:
LW - L - 2W + 2 = 52
LW - L - 2W - 50 = 0
To find a function in terms of the variable x, we can introduce a new variable x = L - W. Substituting x for L - W, we get:
(L - W)W - (L - W) - 2W - 50 = 0
xW - x - 2W - 50 = 0
(x - 2)W - (x + 50) = 0
Now, we have the area A expressed in terms of the variable x:
A = (x + 50)/(x - 2)
The function A(x) = (x + 50)/(x - 2) represents the area of the printed region in square inches as a function of the variable x.