Polynomial and rational functions

Question

Creative Candles is designing a rectangular gold plaque to hang in their show room. A rectangular etched region of the plaque x inches wide and y inches high must have an area of 88 square inches. There will be 2-inch borders at the top and bottom and 1-inch borders at the sides of the etched region.

A) Write a function for the area A of the plaque. Find the domain of the area function. Then suggest a reasonable domain for the width x of the etched region. Explain.

B) Find the asymptotes of the graph of A(x).

C) What should the dimensions of the plaque be so that it covers the least amount of space on the wall?

Answer 1

The area of the etched region is 88 in² therefore x*y = 88

The plaque however, has a width of x + 1 + 1 and a height of y + 2 + 2, because the plaque includes the borders. Therefore, the area of the plaque is A(x,y) = (x+2)(y+4)

We want this function just in terms of x. Solve for y in the conditional equation

y = 88/x

Then plug that into the area equation

A(x) = (x+2)(88/x + 4) = 4x + 96 + 176/x

Notice that x can't be less than or equal to 0 since there has to be some width. Therefore 0 < x.

Asymptotes exist at x = 0 and y = 4x + 96

For the last part, minimize A(x). Take it's derivative and set it equal to 0

4 - 176/x² = 0

4 = 176/x²

x² = 176/4 = 44

x = √44

Now plug this back in to A(x) to get the min