The maximum area that can be enclosed is 800 square feet.

In summary, an area optimization problem involves finding the maximum or minimum value of a function within a given area or region, while considering constraints. Real-world applications include efficient floor plans, crop yields, and minimizing material waste. The steps to solve such a problem include defining the objective function, identifying constraints, finding critical points, and evaluating the function. The choice of constraints greatly impacts the solution, and common techniques used to solve these problems include Lagrange multipliers, linear programming, and dynamic programming.
  • #1
ksle82
30
0
Given 80 feet of fencing, what is the maximum area that you can enclose along a wall?

Solution:
L=lenght, W=width, A=area

2L+2W=80 (perimeter) ==> L+W=40 ==> L=40-W

LW=A ==> (40-W)(W)= -W^2+ 40W= A

(dA/dW)=0=-2W+40=0 ==> W=20

W=20, L=20--------------------------------------------------ANSWERS

Is this right? any input would be appreciated.
 
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  • #2
I would think only three sides would be fence and the other would be the wall.
 
  • #3
Yea, he's right only 3 sides.
 
  • #4
Doh!
here's the updated solutions:

L+2W=80 ==> L=80-2W

A=LW= (80-2W)(W)= 80W-2W^2

(dA/dW)=-4W+80=0 ==> W=20, L=40 ---------------------Answers
 

FAQ: The maximum area that can be enclosed is 800 square feet.

What is an area optimization problem?

An area optimization problem is a mathematical problem in which the objective is to find the maximum or minimum value of a function, subject to certain constraints, within a given area or region.

What are some real-world applications of area optimization?

Some examples of real-world applications of area optimization include designing efficient floor plans for buildings, optimizing crop yields in agriculture, and minimizing material waste in manufacturing processes.

What are the steps involved in solving an area optimization problem?

The steps involved in solving an area optimization problem typically include defining the objective function, identifying the constraints, finding the critical points, and evaluating the function at those points to determine the maximum or minimum value.

How does the choice of constraints affect the solution to an area optimization problem?

The choice of constraints can greatly impact the solution to an area optimization problem. Different constraints can result in different feasible regions, and therefore, different optimal solutions.

What are some common techniques used to solve area optimization problems?

Some common techniques used to solve area optimization problems include the method of Lagrange multipliers, linear programming, and dynamic programming. Each of these techniques has its own advantages and is suitable for different types of optimization problems.

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