Optimization problem involving an area and perimeter

In summary: Yes, I can. \frac{a}{\frac{b}{c}} = \frac{b}{c} - \frac{a}{\frac{b}{c}}In summary, Apostol's Calculus Vol. 1, ed.2 has a problem that my friend and I have been unable to solve. We are currently stuck and looking for help. We wonder if the problem might be in the setup or if the book is giving us a simplified solution that we are not able to work with. We also tried simplifying the answer but could not find a way to do so that followed the book's solution.
  • #1
duncan idaho
2
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My friend and I have come across this problem in Apostol's Calculus Vol. 1, ed.2 (exercises 4.21 if anyone is looking). We are studying calculus independently and have become stumped by this one.

Homework Statement



The problem as written in Apostol:
"A farmer wishes to enclose a rectangular pasture of area A adjacent to a long stone wall. What dimensions require the least amount of fencing?"

Homework Equations


From the problem setup, we have deduced two elementary equations:

P = 2x + y
A = xy

where P is the perimeter (or length of fence, minus the stone wall part) and A is area (a constant).

The Attempt at a Solution



There are two equations here and two variables. That seemed like a good thing. We found y = A/x and then substituted into the perimeter equation: P(x) = 2x + A/x.

In order to optimize, we differentiated P(x):

P`(x) = 2 - (A/x^2)

Then set the equation to 0 find the extrema and to solve for the variable x.

0 = 2 - (A/x^2)

2 = A/(x^2)

x^2 = A/2

x = (A/2)^1/2

And then substituted x into the area equation to find y:

y((A/2)^1/2) = A

y = (A/(A/2)^1/2))

According to the answer key this is wrong and we've been unable to find our error. It seems like something in the setup is wrong but we've reached a block. Does anyone have any insight into this?

Thank you in advance!
 
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  • #2
Welcome to PF!

Hm, everything looks okay as far as I can tell. Did you try simplifying your answer? I wonder if the book isn't just listing this simplified form since
[tex]
\frac{A}{\sqrt{\frac{A}{2}}}
[/tex]
can be simplified.
 
  • #3
I should have given the answer the book presented. Here it is:

x = ½√(2A)
y = √(2A)

I don't see how the solutions we got could be simplified into the solutions the book gives, however playing algebraically with roots is not my strongest skill. If this is just an algebraic situation I apologize in advance, but if anyone has some insight into this it is appreciated; I just want to make sure that our approach is sound.
 
  • #4
Try using some of the properties listed at http://en.wikipedia.org/wiki/Square_root#Properties to help simplify your answer. That should help you get the answer listed in the book. And in general, can you simplify
[tex]
\frac{a}{\frac{b}{c}} \; ?
[/tex]
 

FAQ: Optimization problem involving an area and perimeter

1. What is an optimization problem involving an area and perimeter?

An optimization problem involving an area and perimeter is a mathematical problem where the goal is to find the dimensions of a shape that will maximize or minimize a given quantity, such as the area or perimeter. This type of problem is commonly encountered in the fields of engineering, physics, and economics.

2. How do you set up an optimization problem involving an area and perimeter?

To set up an optimization problem involving an area and perimeter, you first need to define the variables involved, such as the length, width, and height of the shape. Then, you need to establish an equation that relates these variables to the given quantity that needs to be optimized. Finally, you can use mathematical techniques, such as differentiation, to solve the problem and find the optimal dimensions.

3. What are some real-life examples of optimization problems involving an area and perimeter?

Real-life examples of optimization problems involving an area and perimeter include finding the dimensions of a rectangular garden that will yield the maximum area, determining the dimensions of a box that will have the minimum surface area for a given volume, and maximizing the area of a rectangular field with a fixed amount of fencing.

4. How do you know if you have found the optimal solution in an optimization problem involving an area and perimeter?

In an optimization problem involving an area and perimeter, the optimal solution is typically the point where the derivative of the given quantity with respect to the variable being optimized is equal to zero. This means that the slope of the curve representing the given quantity is flat at this point, indicating that it cannot be increased or decreased any further.

5. What are some challenges that may arise when solving an optimization problem involving an area and perimeter?

Solving an optimization problem involving an area and perimeter can be challenging as it requires a good understanding of mathematical concepts such as calculus and algebra. Additionally, it may be difficult to determine which variable to optimize for, and there may be multiple solutions that need to be evaluated to determine the optimal one. Real-life constraints, such as material limitations, may also add complexity to the problem.

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