Question

Suppose you have 52 feet of fencing to enclose a rectangular dog pen. The function A=26x-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.

Answer 1

To find the width that gives the maximum area, we need to maximize the function A=26x-x^2. One way to do this is by finding the vertex of the parabola represented by the function.

The vertex of a parabola given by the equation y=ax^2+bx+c is given by the x-coordinate:
x = -b/(2a)

In this case, a = -1, b = 26, and c = 0. Plugging these values into the formula for the x-coordinate of the vertex, we have:
x = -26/(2(-1))
x = 13

So, the width that gives the maximum area is 13 feet.

To find the maximum area, we can substitute the value of x into the area function:
A = 26x - x^2
A = 26(13) - 13^2
A = 338 - 169
A = 169

Therefore, the maximum area of the dog pen is 169 square feet.