Question

A rectangular page is to contain 8 square inches of print. The margins at the top and bottom of the page are to be 2 inches wide. The margins on each side are to be 1 inch wide. Find the dimensions of the page that will minimize the amount of paper used. (Let x represent the width of the page and let y represent the height.)

Answer 1

To minimize the amount of paper used, we need to find the dimensions of the page. By setting up equations with the width and height of the page and the margins, we can express the total area of the page in terms of one variable. Using calculus, we can find the minimum value of the total area to determine the dimensions of the page that minimize paper usage.

Step-by-step explanation:

To find the dimensions of the page that will minimize the amount of paper used, we need to consider the area of the page. The total area of the page can be calculated by subtracting the area of the margins from the total area of the page. Let's denote the width of the page as x and the height as y. The total area of the page is xy, and the area of the margins is (2x + 1)(2y + 2).

Given that the page is rectangular and contains 8 square inches of print, we can set up the equation (x - 2)(y - 4) = 8. From this equation, we get xy - 4x - 2y + 8 = 8, which simplifies to xy - 4x - 2y = 0.

We need to find the dimensions that minimize the amount of paper used, which means we need to minimize the total area. To do this, we can express the total area in terms of one variable and use calculus to find the minimum value. We can rewrite the equation xy - 4x - 2y = 0 as y = (4x)/(x - 2). Substituting this expression for y into the equation for the total area, we have A = x(4x)/(x - 2) = (4x²)/(x - 2).

To find the minimum value of A, we can take the derivative of A with respect to x and set it equal to zero. Differentiating A using the quotient rule, we get dA/dx = (8x(x - 2) - 4x^2)/(x - 2)² = 0. Simplifying this equation, we have 4x² - 8x(x - 2) = 0, which further simplifies to x(x - 4) = 0. This equation has two solutions: x = 0 and x = 4. Since the width cannot be zero, the only valid solution is x = 4. Substituting this value back into the equation y = (4x)/(x - 2), we get y = 8.

Therefore, the dimensions of the page that will minimize the amount of paper used are 4 inches by 8 inches.

Answer 2

l = 4 in d = 6 in Note : l represent the width and d represent the height

Step-by-step explanation:

Lets asume the dimensions of a rectangular page is l * d where l is the wide and d is top-bottom leght

So total area of the page is A = l*d ⇒ d = A/l ⇒ d = 8/l

Then the print area of the page will be

l = x + 2 (wide) d = y + 4 (vertical lenght)

So area of the page is

A(x) = (x + 2 ) * ( y + 4 ) but y = 8/l and l= x +2 ⇒ y = 8 ÷ ( x + 2 )

A (x) = ( x + 2) * ( 8 / x + 4 ) ⇒ A(x) = 8 + 4x +16/x + 8 ⇒A(x) = 4x + 16 /x

Taken derivative we have:

A´(x) = 4 + (-1 *(16)/x² ⇒ A´(x) = 4 - 16/x²

A ´(x) = 0 means 4 - 16 /x² = 0 ⇒ 4 x² - 16 = 0 x² = 4 x = 2

Therefore y = 8 ÷ ( x +2) and y = 2

And the dimensions of the page is

l = 4 in d = 6 in