A page should have perimeter of 42 inches. The printing area within the page would be determined by top and bottom margins of 1 inch from each side, and the left and right margins of 1.5 inches from each side. What should be the overall dimensions of the page in order to maximize the printing area?
let the width of the page be x inches
and its length be y inches
2(x+y) = 42
x+y = 21
y = 21-x
width of printed area = x-3
length of printed area = y-2
area = (x-3)(y-2)
= (x-3)(21-x - 2)
= (x-3)(19-x)
= -x^2 + 22x - 57
d(area)/dx = -2x + 22
= 0 for a max of area
2x=22
x = 11
then y = 21-11 = 10
page should be 11 inches wide and 10 inches long
To maximize the printing area, we need to minimize the margins.
Let's assume that the width of the printing area is represented by x inches.
According to the given information, the left and right margins are 1.5 inches each, so the width of the page would be x + 1.5 + 1.5 = x + 3 inches.
Similarly, the top and bottom margins are 1 inch each, so the height of the page would be x + 1 + 1 = x + 2 inches.
The perimeter of the page is the sum of all sides, which can be calculated as follows:
Perimeter = 2(width) + 2(height)
Given that the perimeter should be 42 inches, we can write the equation as:
42 = 2(x + 3) + 2(x + 2)
Now, let's solve for x:
42 = 2x + 6 + 2x + 4
Combine like terms:
42 = 4x + 10
Subtract 10 from both sides:
32 = 4x
Divide both sides by 4:
8 = x
Therefore, the width of the printing area (x) should be 8 inches.
To find the overall dimensions of the page, we can substitute the value of x back into our earlier expressions:
Width = x + 3 = 8 + 3 = 11 inches
Height = x + 2 = 8 + 2 = 10 inches
Thus, the overall dimensions of the page should be 11 inches by 10 inches to maximize the printing area.
To maximize the printing area, we need to determine the dimensions of the page by subtracting the margins from the overall dimensions.
Let's denote the overall width of the page as W and the overall height of the page as H.
Given that the top and bottom margins are each 1 inch, and the left and right margins are each 1.5 inches, we can set up the following equations:
Width of the printing area = W - (1.5 + 1.5) = W - 3 inches
Height of the printing area = H - (1 + 1) = H - 2 inches
The sum of the printing area dimensions should equal the desired perimeter of 42 inches:
2 * (Width of the printing area + Height of the printing area) = 42 inches
Substituting the equations for the printing area dimensions:
2 * (W - 3 + H - 2) = 42 inches
Simplifying the equation:
2W + 2H - 10 = 42
2W + 2H = 52
W + H = 26
Now, we need to maximize the printing area. Since we want to maximize the area, we can treat it as a rectangle and use the formula for the area of a rectangle:
Area of the printing area = (Width of the printing area) * (Height of the printing area)
Area = (W - 3) * (H - 2)
To maximize the area, we can substitute the equation W + H = 26 into the area equation:
Area = (26 - H - 3) * (H - 2)
Area = (23 - H) * (H - 2)
Area = 23H - 2H - 46 - H^2 + 2H
Area = -H^2 + 23H + 2H - H - 46
Area = -H^2 + 24H - 46
To find the maximum area, we can take the derivative of the area equation with respect to H and set it equal to zero:
d(Area)/dH = -2H + 24 = 0
-2H = -24
H = 12
Plugging this value back into the equation W + H = 26:
W + 12 = 26
W = 26 - 12
W = 14
Therefore, the overall dimensions of the page that would maximize the printing area while maintaining a perimeter of 42 inches are 14 inches in width and 12 inches in height.