Calculus optimization

Question

A poster for the animal shelter is on a 7875 cm^2 sheet of glossy paper. The margins on the top and bottom are 2 cm and the side margins are 3 cm. What is the largest possible printable area (in cm^2) of the paper

Answer 1

Hello Precious,

Draw a rectangle and label it L along the bottom and W along the side.

Then the area of the entire poster is

A = L * W = 7875

→ L = 7875 / W = 7875W^-1

Now draw a rectangle within the first rectangle, and label along the bottom as L - 6, and the side as W - 4.

(You have to subtract 3 from each side for the length, and 2 from both top and bottom for the width).

Then the area of the printable area, B, is

B = (L - 6) * (W - 4). Substitute out the L to get

B = (7875W^-1 - 6) * (W - 4)

→ B = 7875 - 31,500W^-1 - 6W + 24

→ B = 7899 - 31,500W^-1 - 6W

Since the printable area is what we want to optimize (maximize in this case), this is the equation we want to differentiate,

→ B' = 31,500W^-2 - 6

→ B' = (31,500 / W²) - 6

→ B' = (31,500 - 6W²) / W²

Now set the numerator equal to zero

→ 31,500 - 6W² = 0

→ 6W² = 31,500

→ W² = 5,250

→ W ≈ 72.457

To ensure this will value will yield the largest possible area, test a value on either side into the derivative.

That is,

B'(70) > 0 and

B'(80) < 0

→ a local max at W ≈ 72.457

→ B_max ≈ 7029.517 cm².

I hope the above helps, and thank you for posting the question.

Michael Ehlers