A rectangular page is to have a print area of 96 square inches. The top and bottom margins are to be 1.5 inches each and the left and right margins are to be 1 inch each. What dimensions will minimize the total area of the page.
I got L=8sqr(2) But I'm not sure how to get (w) or how to continue with the rest of the problem.
If printed area has height y and width x, then the page size is (x+2) by (y+3)
Since xy = 96, the page size
s = (x+2)(y+3) = (x+2)(96/x + 3)
ds/dx = 3 - 192/x^2
so, ds/dx = (3x^2-192)/x^2
which is zero when x=8
so the page is 10x11
So, how do our calculations differ?
oops.The printed area is 8x12, so the page is 10x15
To minimize the total area of the page, we can use the concept of the quadratic function. Let's consider the width of the print area as "w" and the length of the print area as "L".
Given that the top and bottom margins are 1.5 inches each and the left and right margins are 1 inch each, we can calculate the total width and length of the page by adding the margins to the print area:
Total width = w + (1 inch + 1 inch) = w + 2 inches
Total length = L + (1.5 inches + 1.5 inches) = L + 3 inches
The total area of the page can be calculated by multiplying the total width and total length:
Total area = (w + 2 inches) * (L + 3 inches)
However, we know that the print area should have an area of 96 square inches. So we can set up an equation:
w * L = 96 square inches
Now, we need to express L in terms of w so that we can substitute it in the equation for the total area.
From the equation w * L = 96 square inches, we can solve for L:
L = 96 / w
Substituting this value for L in the equation for the total area:
Total area = (w + 2 inches) * (96 / w + 3 inches)
To minimize the total area, we can take the derivative of the total area equation with respect to w and set it equal to zero:
d(Total area) / dw = 0
Differentiating the equation, we get:
(dw + 2) * (96 / w + 3) - (w + 2) * (96 / w^2) = 0
Simplifying the equation, we get:
dw^2 - 96 = 0
Now, we can solve for w by taking the square root of both sides:
w^2 = 96
w = sqrt(96)
Therefore, the width of the print area that minimizes the total area of the page is approximately 9.80 inches.
To find the length, we can substitute this value of w in the equation for L:
L = 96 / w
L = 96 / (sqrt(96))
L = 9.80 inches
Therefore, the dimensions that will minimize the total area of the page are approximately 9.80 inches by 9.80 inches.
To minimize the total area of the page, we need to find the dimensions that minimize the sum of the print area and the margins.
Let's start by assuming the width of the page is w and the length of the page is L.
The print area is given as 96 square inches. Since the margins are 1 inch on the left and right and 1.5 inches on the top and bottom, we can calculate the dimensions of the print area as follows:
Print width = width - 2 * margin width = w - 2(1 inch) = w - 2 inches
Print length = length - 2 * margin length = L - 2(1.5 inches) = L - 3 inches
The area of the print is the product of the print width and print length:
Print Area = (w - 2 inches) * (L - 3 inches) = 96 square inches
To continue solving for the dimensions w and L, we will use the information you provided: L = 8√2.
Substitute L = 8√2 into the equation and solve for w:
(w - 2 inches) * (8√2 - 3 inches) = 96 square inches
Expand the equation:
(w - 2 inches) * 8√2 - (w - 2 inches) * 3 inches = 96 square inches
8√2w - 16√2 inches - 3w + 6 inches = 96 square inches
Combine like terms:
(8√2w - 3w) - 16√2 inches + 6 inches = 96 square inches
(8√2 - 3)w - 16√2 + 6 inches = 96 square inches
(8√2 - 3)w = 96 square inches + 16√2 - 6 inches
(8√2 - 3)w = 96 square inches + 16√2 - 6 inches
Now you can continue solving for w by isolating it on one side of the equation, and then find the value of L using the given L = 8√2.