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

In summary: Another way is to use calculus to find the maximum of A(x) = 40x - 2x^2 by setting the derivative equal to zero and solving for x. The answer from this method should match the answer from the first method. In summary, the farmer wants to build a rectangular pen with a maximum possible area, given a specific amount of fencing material. The maximum area can be found by completing the square or using calculus to find the vertex of the parabola that represents the area function.
  • #1
csc2iffy
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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?
 
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  • #2
csc2iffy said:

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.
 

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

1. How do you approach a maximizing area word problem?

To solve a maximizing area word problem, you need to first identify the given information and the unknown variable. Then, you can use the formula for finding the area of a shape (such as A = length x width for a rectangle) and set up an equation with the given information. Finally, use algebraic techniques to solve for the unknown variable and determine the maximum area.

2. What is the importance of maximizing area in real-world applications?

Maximizing area is important in real-world applications because it allows us to use our resources efficiently. For example, in agriculture, maximizing the area of farmland can result in higher crop yields. In construction, maximizing the area of a building can lead to more livable space. By solving maximizing area word problems, we can make informed decisions to optimize our use of space.

3. Is there a specific strategy for solving maximizing area word problems?

Yes, there are certain strategies that can be helpful in solving maximizing area word problems. Some common strategies include breaking down complex shapes into simpler ones, using geometry principles to find relationships between different variables, and understanding the given constraints to narrow down the possible solutions.

4. How can I check if my solution to a maximizing area word problem is correct?

You can check your solution by plugging it back into the original problem and seeing if it satisfies all the given constraints. Additionally, you can use the second derivative test to confirm that your solution is indeed the maximum area.

5. Are there any tips for solving maximizing area word problems quickly?

One tip for solving maximizing area word problems quickly is to practice regularly and familiarize yourself with different types of word problems. Additionally, it can be helpful to draw diagrams and label all the given information to visualize the problem better. Breaking down the problem into smaller steps and using shortcuts, such as the Pythagorean theorem, can also save time in the solving process.

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