Answer:
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.
<em>Hence, the dimensions of the smallest piece that can be used are: 10 by 20 and the area is 200 square inches</em>