A page of print containing 144 square centimeters of printed region has a margin of 4 and 1/2 centimeters at the top and bottom and a matgin of 2 centimeters at the sides. What are the dimensions of the oage if the width across the page is four units of the length?
To solve this problem, let's start by defining the variables:
Let's assume:
Width of the page = W
Length of the page = L
From the given information, we can deduce the following equations:
1) Area of the printed region = 144 square centimeters
The area of the printed region is the area of the page minus the areas of the margins:
Area of the printed region = (W - 2*2) * (L - 4.5 - 4.5) = (W - 4) * (L - 9)
We know that the area of the printed region is 144 square centimeters:
(W - 4) * (L - 9) = 144
2) The width across the page is four units of the length:
W = 4L
Now, we have two equations, and we can solve them simultaneously to find the values of W and L.
Substituting W = 4L in equation (1):
(4L - 4) * (L - 9) = 144
Expanding the equation:
4L^2 - 40L - 4L + 36 = 144
Simplifying the equation:
4L^2 - 44L - 108 = 0
Dividing the equation by 4:
L^2 - 11L - 27 = 0
Factoring the equation:
(L - 9)(L + 3) = 0
Solving for L, we get two possible solutions:
L1 = 9
L2 = -3
Since length cannot be negative, we discard L2 = -3.
Now, substitute the value of L1 into W = 4L:
W = 4 * 9 = 36
Therefore, the dimensions of the page are:
Width (W) = 36 units
Length (L) = 9 units