Question

Suppose you have 54 feet of fencing to enclose a rectangular dog pen. the function a = 27x – x2 , 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

We are given function for area$A = 27x-x^2$.

where x = width, gives you the area of the dog pen in square feet.

We need to find the width that gives the maximum area.

And maximum area of the rectangular dog pen.

The given function is a quadratic function.

And a quadratic function represents a parabolic shape on the graph.

The highest point of the parabola is the vertex of the parabola.

So, we need to find the vertex of the parabola for the given function.

A = 27x-x^2.

Formula for x-coordinate of the vertex is = -b/2a.

Plugging values of a and b in formula, we get

-27/ 2(-1) = -27/-2 = 13.5

Therefore, 13.5 feet width would give maximum area.

Let us find the maximum area now.

Plugging x=13.5 in given function, we get

$F(13.5) = 27(13.5) - (13.5)^2$

=364.5 - 182.25

= 182.25 square feet ≈ 182.3

Therefore, 13.5 feet width gives you the maximum area and 182.3 square feet is the maximum area.

Answer 2

Answer:

13.5 feet width gives you the maximum area and 182.3 square feet is the maximum area.

Step-by-step explanation: