A piece of paper is to display ~128~ 128 space, 128, space square inches of text. If there are to be one-inch margins on both sides and two-inch margins at the bottom and top, what are the dimensions of the smallest piece of paper (by area) that can be used? Choose 1

The dimensions of the smallest piece that can be used are: 10 by 20 and the area is 200 square inches

Step-by-step explanation:

We have that:

Let the dimension of the paper be x and y;

Such that:

So:

Substitute 128 for Area

Make x the subject

When 1 inch margin is at top and bottom

The length becomes:

When 2 inch margin is at both sides

The width becomes:

The New Area (A) is then calculated as:

Substitute for x

Open Brackets

Collect Like Terms

To calculate the smallest possible value of y, we have to apply calculus.

Different A with respect to y

Set

This gives:

Collect Like Terms

Multiply through by

Divide through by 2

Take square roots of both sides

Recall that:

Recall that the new dimensions are:

So:

To double-check;

Differentiate A'

The above value is:

This means that the calculated values are at minimum.

Hence, the dimensions of the smallest piece that can be used are: 10 by 20 and the area is 200 square inches

f(2x−x2)=−3x2+6x−1

step-by-step explanation:

set up the composite function and evaluate.

f(2x−x2)=−3x2+6x−1

3, 12, 48, 384, 1536, 6144, 24576, 98304

step-by-step explanation:

just keep timing it by 4.