Optimizing Area for a Sport Center with a Rectangular Region and Semicircle Ends

In summary, the problem is to find the dimensions of a rectangular region with a semi-circle at each end in order to maximize the area, given a perimeter of 500 meters. The perimeter formula is given by 2l + 2pi r, and the area formula is lw + pi r^2. Using these formulas, the problem can be solved by finding the stationary points of the area function and checking which one is a maximum point. Alternatively, Lagrange multipliers can be used.
  • #1
david12
5
0
help me out on this proble i am confuse

a sport center is to be constructed.it consists of a rectangular region with a semicircle ach end .if the perimater of the room is to be a 500 meter running truck find the dimetion that will make the area as large as possible.

i can find if the picture only has a rectange.but the perimater of the half circle which is pir^2

p = 2lw + PI r

500=2lw + pi r

i don't know where to go after this please help me out
 
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  • #2


Hmm you can use Lagrange multipliers to solve this but it may be unnecessary. Otherwise you can just find the values of l,r (note that w=2r) such that Area is maximised. You wrote p when it should meant Area instead. So find the stationary points of the function A(r,l) and check which is a max point.
 
  • #3


david12 said:
help me out on this proble i am confuse

a sport center is to be constructed.it consists of a rectangular region with a semicircle ach end .if the perimater of the room is to be a 500 meter running truck find the dimetion that will make the area as large as possible.

i can find if the picture only has a rectange.but the perimater of the half circle which is pir^2

p = 2lw + PI r
No. First, the perimeter of a rectangle is 2l+ 2w, not "2lw" (you are mixing perimeter and area formulas). Also if you run around the half circle you do NOT run along that end of the rectangle. Taking the semi-circle to be on the "w" end, the distance is 2l+ w+ pi r.


500=2lw + pi r
500= 2l+ w+ pi rl. Also, have you drawn a picture of this? Is so you should have been able to see that the semi-circle has one end (w) of the rectangle as diameter: r= 2w.

i don't know where to go after this please help me out
You want to maximize the area. What is the formula for the area, as a function of l and w? You can use the perimeter formula to eliminate one of the variables.

Edit: Does "with a semicircle ach end" mean a semi-circle at each end? That makes more sense! I was wondering about runing around the corners! In that case, there is no "w" length so the perimeter is 2l+ 2pi r. And, of course, since we now have an entire circle added the area is lw+ pi r^2= 2lr+ pi r^2.
 
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  • #4


What does 'lw' mean? The perimeter of a semicircle is pi*r (not pi*r^2), but there are two of them in the perimeter. You'll really need to be a lot clearer about what you are doing. Once you said what all the symbols mean, what's a formula for the area?
 

1. What are Lagrange multipliers and why are they important in optimization?

Lagrange multipliers are a mathematical method used to find the maximum or minimum value of a function subject to a set of constraints. They are important in optimization because they allow us to find the optimal solution to a problem while taking into account the constraints that must be satisfied.

2. How do Lagrange multipliers work?

Lagrange multipliers work by introducing a new variable, called the multiplier, to the objective function. This new variable represents the impact of the constraints on the objective function and allows us to find the optimal solution by taking the gradient of the objective function and setting it equal to the gradient of the constraint function multiplied by the multiplier.

3. What types of problems can be solved using Lagrange multipliers?

Lagrange multipliers can be used to solve both constrained and unconstrained optimization problems. They are commonly used in economics, engineering, and physics to find the optimal solution to a variety of problems, such as maximizing profits or minimizing energy consumption.

4. When should I use Lagrange multipliers?

Lagrange multipliers are most useful when dealing with problems that have multiple constraints and require finding the optimal solution. They are also helpful when the constraints are non-linear or when the objective function is difficult to optimize directly.

5. Are there any limitations to using Lagrange multipliers?

While Lagrange multipliers are a powerful tool for optimization, they do have some limitations. They may not always provide the global optimal solution, and they can be computationally expensive for problems with a large number of constraints. Additionally, they may not be suitable for problems with discontinuous or non-differentiable objective functions.

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