A printed page is to have 1.5 inches margin on all sides. The page should contain 144 square inches of type. What dimension of the page will minimize the area while still meeting these other requirements?
If there is no limit to the dimensions of the page, a square will have minimal area. If the printed area is
12x12, the sheet is 15x15 with area 225
6x24, the sheet is 9x27 with area 243
1x144, the sheet is 4x147 with area 588.
I think you have left out some pertinent constraints.
.1x1440, the sheet is 3.1x1443 with area 4473
To find the dimension of the page that will minimize the area while still meeting the requirements, let's denote the width of the page as "w" and the height of the page as "h".
Based on the given information, the printable area inside the margins will have dimensions of "w - 3 inches" for the width and "h - 3 inches" for the height.
The area of this printable area can be calculated by multiplying the width and height:
Area = (w - 3) * (h - 3)
We are given that the area of the printable type should be 144 square inches, so we can write the equation:
144 = (w - 3) * (h - 3)
To minimize the area, we need to take the derivative of the equation with respect to either w or h and set it equal to zero. However, since we are only interested in finding the dimensions, we can solve for one variable in terms of the other and substitute it into the equation.
Let's solve for h in terms of w:
144 = (w - 3) * (h - 3)
144 = h * w - 3h - 3w + 9
135 = h * w - 3h - 3w
Rearranging the terms:
h * w = 3h + 3w + 135
Solving for h:
h = (3w + 135) / (w + 3)
Now, we can substitute this expression for h into the area equation:
Area = (w - 3) * ((3w + 135) / (w + 3) - 3)
To further simplify, let's multiply both terms by (w + 3) to clear the fraction:
Area = (w - 3)(3w + 135 - 3(w + 3))
Expanding and combining like terms:
Area = (w - 3)(3w + 135 - 3w - 9)
Area = (w - 3)(-6)
Now, we can find the critical points by setting the derivative of the area equation with respect to w equal to zero:
d(Area)/dw = -6 = 0
Solving for w:
-6 = 0
This equation has no solution.
Hence, there is no critical point for this function, which means that the area is always decreasing (or increasing) as w changes.
Therefore, there is no minimum area that satisfies the given requirements.
To solve this problem, we need to find the dimensions of the page that minimize its area while satisfying the given constraints.
Let's assume the width of the page is x inches.
Since there is a 1.5 inch margin on each side, the actual width available for the text would be x - 2(1.5) = x - 3 inches.
Similarly, the height of the page would be y inches (to be determined).
The area of the page is given by width multiplied by height.
Therefore, the area A can be expressed as:
A = (x - 3) * y
Now, we are given that the page should contain 144 square inches of type. This means that the area of the text on the page is 144 square inches.
Considering the margin, the area of the text is given by:
Text area = (x - 3 - 2(1.5)) * (y - 2(1.5)) = (x - 6) * (y - 3)
Since the text area is given as 144 square inches, we have:
(x - 6) * (y - 3) = 144
Now, we need to minimize the area A = (x - 3) * y while still satisfying the constraint (x - 6) * (y - 3) = 144.
To find the dimensions that minimize the area, we need to find the critical points by taking the derivative of the area A with respect to x and y and setting them equal to zero.
Taking the partial derivative of A with respect to x:
∂A/∂x = y - 3 = 0
Taking the partial derivative of A with respect to y:
∂A/∂y = x - 3 = 0
Solving these two equations simultaneously gives x = 3 and y = 3.
Therefore, the dimensions of the page that minimize the area while still meeting the other requirements are 3 inches by 3 inches.