Question

A rectangular page is designed to contain 81 square inches of print. The margins at the top and bottom of the page are each 1 inches deep. The margins on each side are 1½ inches wide. What should the dimensions of the page be so that the least amount of paper is used?

Answer 1

The dimensions of the rectangular  page will be 11.02 inches in width and 7.35 inches in height so that the least amount of paper is used.

Given that,

The area of the rectangular print = 81 square inches

The margins at the top of the page = 1inches, and

The margins at the bottom of the page  = 1inches,

And,

The margins on each side = 1(1/2) inches wide.

Now,

Let, the width of the print = x

And,

The height of the print = y

Now,

According to the question,

The width of the page = x + (1.5 + 1.5) = x + 3

And,

The height of the page = y + (1 + 1) = y + 2

Now,

i.e.

xy = 81

we get,

y = (81/x)

Now,

The area of the page (A) = (x + 3)(y + 2) = (x + 3) [(81/x) + 2],

Now,

For the least amount of paper to be used,

dA/dx = (x + 3) [(81/x) + 2]

We get,

dA/dx = [(81/x) + 2] + (x + 3) [-81/x²] = 0

[(81+2x)/x] = [(x + 3) * 81]/x²        

81 + 2x² = 81x + 243

2x² = 243

We get,

x = 11.02 inch

So,

y = 81/x = 81/11.02 = 7.35 inch

Hence we can say that the dimensions of the rectangular page will be 11.02 inches in width and 7.35 inches in height so that the least amount of paper is used.

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brainly.com/question/10046743

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