Answer:
[tex]Height = 8.660\ in[/tex]
[tex]Width = 17.321\ in[/tex]
Step-by-step explanation:
Given
[tex]Area= 150in^2[/tex]
[tex]Side\ Margin = 1\ in[/tex]
[tex]Top\ \&\ Bottom\ Margins = 2\ in[/tex]
Required
Determine the smallest dimension to use
To answer this question, I'll make use of the attached figure as a point of reference.
The area of the printed matter is:
[tex]Area = Length * Width[/tex]
[tex]A_1 = H * W[/tex]
Substitute 150 for A1
[tex]150 = H * W[/tex]
Make H the subject
[tex]H = \frac{150}{W}[/tex]
The area of the full paper is:
[tex]Area = Length * Width[/tex]
[tex]A_2 = (W + 2+2) * (H + 1 + 1)[/tex]
[tex]A_2 = (W + 4) * (H + 2)[/tex]
Substitute 150/W for H
[tex]A_2 = (W + 4) * (\frac{150}{W} + 2)[/tex]
Open brackets
[tex]A_2 = W(\frac{150}{W} + 2) + 4(\frac{150}{W} + 2)[/tex]
[tex]A_2 = 150 + 2W + \frac{600}{W} + 8[/tex]
Collect Like Terms
[tex]A_2 =2W + \frac{600}{W} + 8+150[/tex]
[tex]A_2 =2W + \frac{600}{W} + 158[/tex]
Differentiate with respect to w and set the result to 0
[tex]A_2' = 2 - \frac{600}{W^2} + 0[/tex]
[tex]A_2' = 2 - \frac{600}{W^2}[/tex]
Set to 0
[tex]0 = 2 - \frac{600}{W^2}[/tex]
Collect Like Terms
[tex]2 = \frac{600}{W^2}[/tex]
Cross Multiply
[tex]2 * W^2 = 600[/tex]
Make [tex]W^2[/tex] the subject
[tex]W^2 = \frac{600}{2}[/tex]
[tex]W^2 = 300[/tex]
Take positive square root of both sides
[tex]W = 17.321[/tex]
Recall that:
[tex]H = \frac{150}{W}[/tex]
[tex]H = \frac{150}{17.321}[/tex]
[tex]H = 8.660[/tex]
Hence, the smallest dimension of the paper is:
[tex]Height = 8.660\ in[/tex]
[tex]Width = 17.321\ in[/tex]
Answer:
H=18
W=12
Step-by-step explanation:
The question says 1 inch margins on the sides AND the top. And THEN a 2 inch margin ONLY on the bottom.
First, solve for y,
A = xy
150 = xy
150/x = y
If the center dimensions is A and its it equals x*y, then plus the margins, the outer dimensions would just be x+(size of margins) for width and y+(size of margins) for height.
This gives us
Width: x + 2
Height: y + 3
Putting height into terms of x: (Substitute y) = 3 + 150/x
Now, since the area of a rectangle is b*h, the equation is:
(x + 2)(3 + 150/x)
Simplify:
3x + 150 + 6 + 300/x
= 3x + 156 + 300/x
Next, we find the derivative of this and then solve for x.
= 3 - 300/x^2
Solving for x:
0 = 3 - 300/x^2
3 = 300/x^2
3x^2 = 300
x^2 = 100
x = 10
Now that we have x, we plug this into the different equations for height and base:
Base=(10) + 2 = 12
Height=3 + 150/(10) =3+15 = 18
Hope this helps!
