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.
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.