Can someone help me with this??? Optimization!!!?

Question

A book designer has decided that the pages of a book should have 1in. margins at the top and bottom and 1/2in. margins on the sides. She further stipulated that each page should have an area of 50 sq. inches. Determine the page dimensions that will result in the maximum printed area on the page.

Answer 1

L*w = 50, so w = 50/L = 50*L^(-1)

We're optimizing the Printed area: P = (L-2)(w-1)

Replacing w,

P = (L-2)(50L^(-1) - 1)
Multiplying:
P = 50 -100L^(-1) - L + 2 = 52 - 100L^(-1) - L

P' = 100L^(-2) - 1
P' = 100/(L^2) - 1 = (100-L^2)/L^2

Set P' = 0, you get L = 10 (or -10, which you disregard)

So the length of the page is 10 in, and the width must be 5 inches to get area of 50 sq. inches

Answer 2

Area of page = 50 sq in
Let its page size be L in x B in
ie LB = 50 so B = 50/L

Thus print area A = (L - 2)(B - 1)
= LB - L - 2B + 2
= 50 - L - 100/L + 2
= 52 - L - 100/L

Now dA/dL = -1 + 100/L²
= 0 for stationary points
So L² = 100
ie L = ±10
Ignoring L<0
L = 10
d²A/dL² = -200/L³ < 0 when L > 0
So the curve is concave down
ie the printing area is maximum when L = 10

So the dimensions of the page need be 10 in x 5 in to maximise the printing area