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:
[tex]Area = 128[/tex]
Let the dimension of the paper be x and y;
Such that:
[tex]Length = x[/tex]
[tex]Width = y[/tex]
So:
[tex]Area = x * y[/tex]
Substitute 128 for Area
[tex]128 = x * y[/tex]
Make x the subject
[tex]x = \frac{128}{y}[/tex]
When 1 inch margin is at top and bottom
The length becomes:
[tex]Length = x + 1 + 1[/tex]
[tex]Length = x + 2[/tex]
When 2 inch margin is at both sides
The width becomes:
[tex]Width = y + 2 + 2[/tex]
[tex]Width = y + 4[/tex]
The New Area (A) is then calculated as:
[tex]A = (x + 2) * (y + 4)[/tex]
Substitute [tex]\frac{128}{y}[/tex] for x
[tex]A = (\frac{128}{y} + 2) * (y + 4)[/tex]
Open Brackets
[tex]A = 128 + \frac{512}{y} + 2y + 8[/tex]
Collect Like Terms
[tex]A = \frac{512}{y} + 2y + 8+128[/tex]
[tex]A = \frac{512}{y} + 2y + 136[/tex]
[tex]A= 512y^{-1} + 2y + 136[/tex]
To calculate the smallest possible value of y, we have to apply calculus.
Different A with respect to y
[tex]A' = -512y^{-2} + 2[/tex]
Set
[tex]A' = 0[/tex]
This gives:
[tex]0 = -512y^{-2} + 2[/tex]
Collect Like Terms
[tex]512y^{-2} = 2[/tex]
Multiply through by [tex]y^2[/tex]
[tex]y^2 * 512y^{-2} = 2 * y^2[/tex]
[tex]512 = 2y^2[/tex]
Divide through by 2
[tex]256=y^2[/tex]
Take square roots of both sides
[tex]\sqrt{256=y^2[/tex]
[tex]16=y[/tex]
[tex]y = 16[/tex]
Recall that:
[tex]x = \frac{128}{y}[/tex]
[tex]x = \frac{128}{16}[/tex]
[tex]x = 8[/tex]
Recall that the new dimensions are:
[tex]Length = x + 2[/tex]
[tex]Width = y + 4[/tex]
So:
[tex]Length = 8 + 2[/tex]
[tex]Length = 10[/tex]
[tex]Width = 16 + 4[/tex]
[tex]Width = 20[/tex]
To double-check;
Differentiate A'
[tex]A' = -512y^{-2} + 2[/tex]
[tex]A" = -2 * -512y^{-3}[/tex]
[tex]A" = 1024y^{-3}[/tex]
[tex]A" = \frac{1024}{y^3}[/tex]
The above value is:
[tex]A" = \frac{1024}{y^3} > 0[/tex]
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
