Some years from now you are working for a book publisher. Your boss asks you to give him a formula that will tell him the length and width of a book page that contains A square inches of printed text, a left margin of L inches, a right margin of R inches, a top margin of T inches, and a bottom margin of B inches, and that otherwise has an area as small as possible. After dusting off your Calculus notes you tell him that the length of that page equals _________ inches, and the width equals _________ inches. Of course, both of your answers are in terms A, L, R, T, and B
To determine the length and width of the book page, we can start by subtracting the margins from each dimension. We'll calculate the length first:
1. Subtract the left and right margins (L + R) from the available width.
2. Divide the remaining width by the desired area of printed text (A) to obtain the effective width per inch.
3. Multiply the effective width per inch by the total area of printed text (A) to determine the width of the page minus the margins.
4. Finally, add the left and right margins (L + R) to find the overall length of the page.
So, the length of the book page would be: (L + R) + (A / ((page width - L - R) / A))
Now, let's calculate the width of the book page:
1. Subtract the top and bottom margins (T + B) from the available height.
2. Divide the remaining height by the desired area of printed text (A) to obtain the effective height per inch.
3. Multiply the effective height per inch by the total area of printed text (A) to determine the height of the page minus the margins.
4. Finally, add the top and bottom margins (T + B) to find the overall width of the page.
Therefore, the width of the book page would be: (T + B) + (A / ((page height - T - B) / A))
Please note that the formulas provided are in terms of A (area of printed text), L (left margin), R (right margin), T (top margin), and B (bottom margin).
To find the length and width of a book page with specific margins and a minimum area, we can start by setting up the problem.
Let's assume the length of the page is represented by 'x' inches, and the width is represented by 'y' inches. We want to minimize the area of the page, given the constraints.
The area of the page can be calculated by subtracting the margins from the total page area. The total page area is the length multiplied by the width, while the margins will reduce the effective printable area.
The effective printable area can be calculated by subtracting the margins from the length and width dimensions. Hence, the effective length is (x - L - R) inches, and the effective width is (y - T - B) inches.
Now, the area of the printed text on the page is A square inches, and we want to minimize the remaining area.
The remaining area is the area of the page minus the area of the printed text. Therefore, the area of the page is xy, and the remaining area is given by:
Remaining Area = xy - A
To find the minimum remaining area, we can take the derivative of the remaining area with respect to x and y, set both derivatives equal to zero, and solve for x and y.
∂(Remaining Area)/∂x = y - A/x = 0 ...(Equation 1)
∂(Remaining Area)/∂y = x - A/y = 0 ...(Equation 2)
Solving Equation 1 for x:
y - A/x = 0
y = A/x
Substituting this value of y in Equation 2:
x - A/(A/x) = 0
x^2 - A = 0
x^2 = A
Taking the square root of both sides:
x = √A
Similarly, substituting the value of x in Equation 1:
y = A/(√A)
y = √A
So, the length of the page is √A inches, and the width of the page is also √A inches.
Therefore, the length of the page equals √A inches, and the width equals √A inches.
Note: It's important to keep in mind that this solution assumes a rectangular page, and the minimum area is achieved by making the page a square.
If the printed area has dimensions x and y, then xy=A, so y=A/x
The actual page has dimensions
x+L+R and y+T+B
To minimize the page area, then you want to minimize
f(x,y) = (x+L+R)(y+T+B)
f(x) = (x+L+R)(A/x + T+B)
= (T+B)x + A + A(L+R)/x + (L+R)(T+B)
df/dx = (T+B) - A(L+R)/x^2
df/dx = 0 when x^2 = A(L+R)/(T+B)
. . .