How to Verify Maximum Area of a Rectangular Pen with Limited Fencing?

[csc2iffy]

Homework Statement

A farmer wants to build a rectangular pen. He has a barn wall 40 feet long, some or all of which must be used for all or part of one side of the pen. In other words, with f feet of of fencing material, he can build a pen of perimeter ≤ f+40, and remember he isn't required to use all 40 feet.
What is the maximum possible area for the pen if:
a. 60 feet of fencing material is available
b.100 feet of fencing material is available
c. 160 feet of fencing material is available

Homework Equations

a.
P=> 2x+y=60 => y=60-2x
A=> xy=60x-2x^2

b.
P=> 2x+y=100 => y=100-2x
A=> xy=100x-2x^2

c.
P=> 2x+y=160 => y=160-2x
A=> xy=160x-2x^2

The Attempt at a Solution

I worked through the problem and found
a. x=15, y=30 => A=450 sq ft
b. x=25, y=50 => A=1250 sq ft
c. x=40, y=80 => A=3200 sq ft

I was just wondering if there was a way I could check these answers?

 

[Mark44]

Homework Statement

A farmer wants to build a rectangular pen. He has a barn wall 40 feet long, some or all of which must be used for all or part of one side of the pen. In other words, with f feet of of fencing material, he can build a pen of perimeter ≤ f+40, and remember he isn't required to use all 40 feet.
What is the maximum possible area for the pen if:
a. 60 feet of fencing material is available
b.100 feet of fencing material is available
c. 160 feet of fencing material is available

Homework Equations

a.
P=> 2x+y=60 => y=60-2x
A=> xy=60x-2x^2

b.
P=> 2x+y=100 => y=100-2x
A=> xy=100x-2x^2

c.
P=> 2x+y=160 => y=160-2x
A=> xy=160x-2x^2

The Attempt at a Solution

I worked through the problem and found
a. x=15, y=30 => A=450 sq ft
b. x=25, y=50 => A=1250 sq ft
c. x=40, y=80 => A=3200 sq ft

I was just wondering if there was a way I could check these answers?

There are at least a couple of ways, one of which doesn't use calculus. In each case your area function, A(x) has a graph that is a parabola that opens downward. The maximum area is attained at the vertex of the parabola. Complete the square to find the vertex.