Answer:
x = 11.02 inch by y = 7.35 inch
Explanation:
Let the width of the print be = x
Then the width of the page is x + (1.5+1.5) = x + 3
Let the height of the print be = y
Then the height of the page is y + (1+1) = y + 2
Given the area of the print is 81 square inches
That means, xy = 81
Therefore, y = [tex]\frac{81}{x}[/tex]
Therefore, the area of the page is
A = (x + 3)(y + 2)
= (x + 3)([tex]\frac{81}{x}[/tex] + 2)
Now for least amount of paper to be used,
[tex]\frac{dA}{dx}=(x+3)(\frac{81}{x}+2)[/tex]
[tex]\frac{dA}{dx}= \left (\frac{81}{x}+2 \right )+\left ( x+3 \right )\left ( -\frac{81}{x^{2}} \right )=0[/tex]
[tex]\frac{81+2x}{x}=\frac{(x+3)(81)}{x^{2}}[/tex]
[tex]81x+2x^{2}=81x+81\times 3[/tex]
[tex]2x^{2}=243[/tex]
x = 11.02 inch
Now y = [tex]\frac{81}{x}[/tex]
= [tex]\frac{81}{11.02}[/tex]
= 7.35 inch
Therefore the dimensions of thr page for the least amount of paper is x = 11.02 inch by y = 7.35 inch
