Suppose you have 80 ft of fence to enclose a rectangular garden. A=40x-x^2 gives you the area of the garden in square feet where x is the width in feet.

Question

What width gives you the maximum gardening area? Please explain how you were able to solve this. Include an equation

Answer 1

The area A is in the form of a polynomial.  We can rewrite the polynomial in which the degree of the terms are in descending order.

 

A = -x² + 40x

 

A is a parabola that opens downward.  This will indicate a graph with a maximum value.  To find the maximum value, we can put A in vertex form.

 

A = a(x - h)² + k

 

where:

a = the coefficient of x² term

 

vertex has coordinates (h, k).

 

A = -(x² - 40x + 400) + 400

A = -(x - 20)² + 400

 

The vertex is  (20, 400). 

 

 

The width that gives you the maximum gardening area is 20 ft.  This is less than the 80 ft of fencing you have to cover the area, so you will have some fencing left over.