Optimizing Dimensions and Cost in Golf Net and Fencing Projects

In summary, the first question involves finding the dimensions of a rectangular prismic net enclosure that will minimize the amount of netting needed and have a volume of 144 m3. The second question involves finding the dimensions of a rectangular lot that can be fenced for a cost of $9000 using two different types of fencing. To solve these types of problems, it is helpful to write down the quantity to be minimized or maximized as a function and use other facts from the problem to simplify the expression until it is in terms of one variable. Calculus techniques can then be used to find the optimal solution. Drawing a diagram can also be helpful in visualizing the problem.
  • #1
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Homework Statement



Two related type of questions:

1) A rectangular prismic net enclosure for practising golf shots is open at one end. Find the dimensions that will minimize the amount of netting needed and give a volume of 144 m3. Netting is only required on the sides, top, and the far end. Height is x, width is also x, and length is y.

2) A rectangular piece of land is to be fenced using two kinds of fencing. Two opposite sides will be fenced using $6/m fencing, while the other two sided will require $9/m fencing. What are the dimensions of the rectangular lot of greatest area that can be fenced for a cost of $9000?

Homework Equations



A'(x) = 0 for max/min

The Attempt at a Solution



In both questions, I don't know what to do with the 144m3 or the $9000.

What equations am I supposed to use? I could also use tips on how to make equations for these types of questions.
 
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  • #2
Tips:

When doing maximization or minimization problems, you will want to write down the quantity to be minimized or maximized as a function, like f(x,...) = <whatever>. Then you will want to use the other facts in the problem to help simplify the expression for f(x,...) until it is an expression in just one independent variable (often called x, although it can really be anything). Then you take the derivative of the function, set it to zero, and do the usual exploration of the endpoints of the interval and the critical points.

So in the net problem, what is supposed to be maximized or minimized? Can you write down an expression for it in terms of the variables in the problem?

Once you've done that, how many variables are there that f depends on? Can you write another equation involving facts from the problem and two of the variables that f depends on in order to eliminate one of them from the f expression? Then you can do your usual calculus stuff and find the answer.
 
  • #3
I find drawing a diagram helps too.
 
  • #4
thanks.
 

FAQ: Optimizing Dimensions and Cost in Golf Net and Fencing Projects

1. What is optimization?

Optimization is the process of finding the best possible solution for a given problem. It involves maximizing or minimizing a certain objective while taking into consideration any constraints or limitations.

2. How is optimization used in science?

Optimization is used in various fields of science, such as physics, engineering, and biology, to name a few. It is used to find the most efficient or effective solution to complex problems, such as designing structures, optimizing processes, or understanding biological systems.

3. What are some common methods used for optimization?

Some common methods used for optimization include linear programming, genetic algorithms, simulated annealing, and gradient descent. These methods use mathematical and computational techniques to find the optimal solution.

4. Can optimization be applied to real-world problems?

Yes, optimization is commonly applied to real-world problems in various industries and fields. It can be used to improve efficiency, reduce costs, or find the best solution for complex systems.

5. What are some challenges in optimization?

One of the main challenges in optimization is dealing with complex and large-scale problems where finding the optimal solution is computationally expensive. Another challenge is balancing multiple objectives or constraints, which may require trade-offs to be made.

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