Question

Suppose you have 54 feet of fencing to enclose a rectangular dog pen. The function A=27x-x^2 , where x = width, gives you the area of the dog pen in square feet. What width gives you the maximum area? What is the maximum area? Round to the nearest tenth as necessary.

width = 13.5 ft; area = 182.3 ft^2

width = 27 ft; area = 182.3 ft^2

width = 13.5 ft; area = 546.8 ft^2

width = 27 ft; area = 391.5 ft^2

Answer 1

Given function: A=27x-x^2 , where x = width, gives you the area of the dog pen in square feet.

The given function is a quadratic function.

And a quadratic function represents a parabola.

The top most point of the parabola is called vertex.

So, we need to find the x-coordinate of the vertex to find the width that gives you the maximum area. Because x represents the width of the rectangle.

A=27x-x^2 could be written in standard form A= -x^2+27x.

We know formula for x-coordinate of the vertex = -b/2a.

For the given function A= -x^2+27x, a=-1 and b=27.

Plugging values in the formula, we get

x-coordinate of the vertex = -b/2a = -27/2(-1) = 27/2 = 13.5 feet.

Plugging x=13.5 in given function, we get

A= -(13.5)^2+27(13.5)

= -182.25 +364.5 = 182.25 ≈ 182.3.

Therefore, width = 13.5 ft gives you the maximum area and the maximum is  area = 182.3 ft^2.