SOLUTION: a. Write a quadratic function to represent to area of all rectangles with a perimeter of 36 feet. b. What is the vertex? c. How can the graph of your function and the vertex

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: a. Write a quadratic function to represent to area of all rectangles with a perimeter of 36 feet. b. What is the vertex? c. How can the graph of your function and the vertex       Log On


   

Question 930356: a. Write a quadratic function to represent to area of all rectangles with a perimeter of 36 feet.
b. What is the vertex?
c. How can the graph of your function and the vertex help you determine which rectangle has the area? What are the side lengths for the rectangle with the greatest area?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
P=2%28L%2BW%29=36
L%2BW=18
L=18-W
.
.
.
A=L%2AW
A=%2818-W%29W
A=18W-W%5E2
A=-%28W%5E2-18W%2B81%29%2B81
A=-%28W-9%29%5E2%2B81
The vertex is (9,81).
.
.
.
The graph isn't really required only the vertex form of the equation.
Since there is a -1 multiplier in front of the quadratic term, the parabola opens downwards so the vertex value is the maximum.
So you know the width (W=9) at the maximum area (A=81).
Then you also know the length L=18-W=18-9=9.
So the maximum area rectangle is a square.