the printed matter on a 10 by 16 centimeter page of a book must cover 40 square centimeters. if all margins are to be the same width, how wide should they be?
let the margin be x cm wide all around
then length of printed area = 16-2x
width of printed area = 10-2x
(16-2x)(10-2x) = 40
160 -52x + 4x^2 = 40
4x^2 - 52x + 120=0
x^2 - 13x + 30 = 0
(x-4)(x-9) = 0
x = 4 or x = 9, but x=9 would produce a negative length and width, so reject it
The margin should be 4 cm wide
(10-2x) (16-2x ) = 40
160-20x -32x + 4x^2 = 40
160- 52x+ 4x^2 = 40
4x^2 -52x + 160 - 40 = 0
4x^2 -52x + 120=0
4(x-10)(x-3) = 0
x = 10, 3
Check
(10-2(3))(16-2(3)) = 40
(10-6)(16-6) = 40
(4)(10) = 40
40 = 40
(10-2(10))(16-2(10) = 40
(10-20)(16-20) = 40
(-10)(-4) = 40
40 = 40
To solve this problem, we can start by finding the total area of the page, including the margin. Then, we subtract the area of the printed matter from the total area to determine the area of the margin. Finally, since all margins have the same width, we divide the area of the margin by the number of margins to find the width of each margin.
Step 1: Find the total area of the page.
The page is 10 by 16 centimeters, which means its total area is 10 * 16 = 160 square centimeters.
Step 2: Subtract the area of the printed matter.
The area of the printed matter is given as 40 square centimeters. So, we subtract 40 from the total area: 160 - 40 = 120 square centimeters.
Step 3: Determine the area of each margin.
Since all margins have the same width, we will divide the total area of the margins by the number of margins. However, we need to figure out how many margins there are.
Considering the width of the entire page (16 centimeters) and the width of the printed matter (unknown), we can deduce that the sum of the printed matter width and the total width of the margins equals the page width (16 centimeters). Therefore, we have: printed matter width + (2 * margin width) = page width.
Since we are looking for the width of each individual margin, we can solve for just the margin width. Rearranging the equation, we get: margin width = (page width - printed matter width) / 2.
Plugging in the values we know:
Margin width = (16 - printed matter width) / 2.
Step 4: Calculate the width of each margin.
Since we have the area of the margin (120 square centimeters) and the width of the margins, we can relate the two by using the formula: area of each margin = width of each margin * width of each margin.
Substituting the variables, we obtain the equation: 120 = margin width * margin width.
Rearranging the equation, we get: margin width = √(120).
Calculating the square root of 120: margin width ≈ 10.95 centimeters.
Therefore, each margin should be approximately 10.95 centimeters wide to ensure that the printed matter on a 10 by 16 centimeter page of a book covers 40 square centimeters and all margins have the same width.