A large rectangular picture frame is to have an area of 120 sq. inches, with 2-inch margins at the top and sides and 1-inch margin at the bottom. What dimensions of the frame...

Question

A large rectangular picture frame is to have an area of 120 sq. inches,

with 2-inch margins at the top and sides and 1-inch margin at the

bottom. What dimensions of the frame will give the largest area for

the picture inside?

Answer 1

Let the dimensions of the frame be p x q.

Then the dimensions of picture inside are (p-4) x (q-3).

Using the constraint p x q = 120, we can express the area of the picture as a function of p alone:

A(p) = (p-4)(120/p-3) = 120 - 3p - 480/p + 12

In general, the area will be maximized either at the boundary of the domain of p,

or at a local maximum.

At the boundary of the domain the area is 0, therefore we just need the consider the

values of p where the derivative A'(p) = 0.

A'(p) = -3 + 480/p² = 0

p = √(480/3) = 12.65

q = 120/p = 9.5