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?
length ---- x
width ----- y
2x+2y = 42
x+y = 21
y = 21 - x
printing area:
length = x-2
width = y-3 = 21-x - 3 = 18-x
area
= (x-2)(18-x)
= 18x - x^2 - 36 + 2x
= -x^2 + 20x - 36
since this is a downwards parabola, the vertex will be the answer to your question.
Find the vertex by whatever mean you learned.
To determine the overall dimensions of the page that would maximize the printing area, we need to subtract the margins from the total perimeter of 42 inches.
Let's start by calculating the total length of the margins. Since there are top and bottom margins of 1 inch each, and left and right margins of 1.5 inches each, we have the following:
Total length of top and bottom margins = 1 inch + 1 inch = 2 inches
Total length of left and right margins = 1.5 inches + 1.5 inches = 3 inches
Therefore, the total length of all four margins combined is 2 inches + 3 inches = 5 inches.
Now, to maximize the printing area, we need to subtract the length of the margins from the total perimeter of the page. So, the printing area's perimeter would be:
Printing area's perimeter = Total perimeter - Total length of margins
Printing area's perimeter = 42 inches - 5 inches = 37 inches
Since the printing area is rectangular, the dimensions of the page would be divided into two equal sides. Let's call this length x. So, the two sides would be:
Length of the printing area = Width of the printing area = x
Now we have the following equation to find x:
2x + 2x = 37
4x = 37
x = 37/4
x = 9.25 inches
So, each side of the printing area should be 9.25 inches, and since the page's dimensions would be equal, the overall dimensions of the page to maximize the printing area would be 9.25 inches by 9.25 inches.