A rectangular page is to contain 97 square inches of print. The margins on each side are 1 inch. Find the dimensions of the page such that the least amount of paper is used.
Let the printed area have dimensions x and y. Then
xy = 97
Now you want the page to have minimum area A, so
A = (x+2)(y+2) = (x+2)(97/x + 2) = 2x+101+194/x
dA/dx = 2-194/x^2
So, find where dA/dx=0
To find the dimensions of the page that use the least amount of paper, we need to minimize the total area of the page.
Let's assume the width of the page is x inches. The length of the page is then given by 97/x since the total area is the product of the width and length.
The dimensions of the printed area are (x - 2) inches in width and (97/x - 2) inches in length, as each side has a 1-inch margin.
The total area of the page, including the margins, is the product of the width and length:
Total Area = (x - 2) * (97/x - 2)
To minimize the area, we can take the derivative of the total area with respect to x and set it equal to zero:
d(Total Area)/dx = 0
Using the product rule, the derivative is:
[x * (d(97/x - 2)/dx) + (x - 2) * (d(x)/dx)] = 0
Simplifying the derivative:
[97 + (2 * x) - 2] / x^2 = 0
97 + 2x - 2x - 4 = 0
97 - 4 = 2x - 2x
93 = 0
We have reached a contradiction, which means there is no value of x that satisfies this equation. This implies that there is no minimum area, and there is no dimension that produces a minimum amount of paper.
To find the dimensions of the page such that the least amount of paper is used, we need to minimize the area of the page while considering the given constraints.
Let's assume the length of the page is L inches, and the width is W inches.
Given that the margins on each side are 1 inch, we can express the length in terms of the width as:
L = W + 2 (1 inch margin on each side)
The total area of the page is the area for printing plus the area of the margins, which can be expressed as:
Area = (L - 2) * (W - 2) + 4
Where (L - 2) * (W - 2) is the area for printing (as the margins on each side reduce the length and width by 2 inches), and 4 represents the area of the four margins.
We are given that the area for printing is 97 square inches. Therefore, we can write the equation as:
(L - 2) * (W - 2) = 97
To minimize the amount of paper used, we need to minimize the area. So, our goal is to minimize the function:
Area = (L - 2) * (W - 2) + 4
To find the dimensions that minimize the area, we can use various methods like calculus or optimization techniques. However, in this case, we can try different values of W and solve for L.
Assume W = 10 inches:
Substituting W = 10 in the equation (L - 2) * (W - 2) = 97, we get:
(L - 2) * 8 = 97
Simplifying further, we find L = 14.125 inches.
Similarly, assume W = 11 inches:
Substituting W = 11 in the equation (L - 2) * (W - 2) = 97, we get:
(L - 2) * 9 = 97
Simplifying further, we find L = 12.889 inches.
Continuing this process, we can find other possible dimensions and calculate the corresponding areas. Ultimately, we will find the dimensions that minimize the area and fulfill the given constraints.
Alternatively, we can use a graphing calculator or software to graph the function Area = (L - 2) * (W - 2) + 4, and find the minimum point on the graph. The coordinates of that point will give us the dimensions of the page that utilize the least amount of paper.