# Scale Models (Sprinklers)

#### Karim

##### New member
I was recently given the question; I hope you won't mind solving this:
A crop field measures 85x75 m
You can use any quantity and combination of sprinklers.
However, there are only three types
Sprinkler 1 has a radial range of 25m
Sprinkler 2 has a radial range of 20m
Sprinkler 3 has a radial range of 15m

A pipe costs \$40 per metre, fully installed. The sprinklers must be connected to a water pipe that runs along the width of the field

Determine the best way to configure the sprinklers and find the cost of installing the irrigation and piping.
Design a scale model to test different combinations and the types of given sizes of water sprinklers.

THE SPRINKLER'S RADIALS MUST NOT OVERLAP AND MUST STAY WITHIN THE CROP FILED

Thank you!

#### Jomo

##### Elite Member
Hi. Yes we do mind solving this. This is a math help forum where we help students solve problems that they are having trouble with. If you tell us what you tried and where you are stuck then we would help you. Thanks.

#### Harry_the_cat

##### Senior Member
This sounds to me like an alternative assessment item ie a maths assignment, not an exam. The OP is looking for an easy way out. Not getting it from me!

#### Karim

##### New member
Hi. Yes we do mind solving this. This is a math help forum where we help students solve problems that they are having trouble with. If you tell us what you tried and where you are stuck then we would help you. Thanks.
Yes well, I did try multiple different methods to solving this. I'm currently using GeoGebra in order to solve this problem, as using a normal graph would be rather difficult for a question such as this. As of right now, I'm using the trial and error method...since I have not found any other way of solving this. Furthermore, in order for me to find out, the most area covered I add the areas of each sprinkler and in order for me to find out the area remaining(not covered by the sprinklers) I would minus the area of the whole plot by the area of which the sprinklers are covering. However, this method of mine is taking some time. Do you know of any other method apart from the "trial and error" method I can use?

#### Karim

##### New member

Hi. Yes we do mind solving this. This is a math help forum where we help students solve problems that they are having trouble with. If you tell us what you tried and where you are stuck then we would help you. Thanks.
Closest I've gotten

#### Karim

##### New member
Hi. Yes we do mind solving this. This is a math help forum where we help students solve problems that they are having trouble with. If you tell us what you tried and where you are stuck then we would help you. Thanks.
Btw I accidentally marked solved...I still haven't sovled it

#### MarkFL

##### Super Moderator
Staff member
Since the cost of pipe is given, I am assuming your objective here is to minimize the cost while maximizing the coverage of the sprinklers. When you say "The sprinklers must be connected to a water pipe that runs along the width of the field" does that mean there must be pipe running along two adjacent sides of the field as your diagram seems to indicate?

I personally wouldn't know how to approach this problem other than by trial and error.

Btw I accidentally marked solved...I still haven't sovled it
I've marked this thread as unsolved.

#### Dr.Peterson

##### Elite Member
If you had explained the context, you might have received more useful suggestions. What techniques have you learned? What knowledge or skill is this assignment intended to test? Is it as entirely open as your wording implies? (The fact that it is an open-ended project is one reason we're not interested in actually solving it, as you asked.)

I would expect it to be largely a trial and error solution, though various ideas might be used to help speed it up. It is reminiscent of various "circle packing" problems, which are generally "solved" by computer algorithms. Are you allowed (or expected) to write a program? Have you learned anything about algorithms, such as "greedy algorithms", to use in such a search?

Of course, yours is far more than a mere circle-packing problem, as you are presumably supposed to minimize cost (which I suppose can be taken to be proportional to the sum of the x-coordinates of the centers, plus the greatest y-coordinate) while also maximizing the total area of the circles, and you have three specified radii available.

The way to start, I think, would be to decide exactly what your goal is (since there are two competing requirements), and how you can efficiently compare two arrangements with respect to that goal; then you can think about how to search for solutions.

#### Karim

##### New member
If you had explained the context, you might have received more useful suggestions. What techniques have you learned? What knowledge or skill is this assignment intended to test? Is it as entirely open as your wording implies? (The fact that it is an open-ended project is one reason we're not interested in actually solving it, as you asked.)

I would expect it to be largely a trial and error solution, through various ideas might be used to help speed it up. It is reminiscent of various "circle packing" problems, which are generally "solved" by computer algorithms. Are you allowed (or expected) to write a program? Have you learned anything about algorithms, such as "greedy algorithms", to use in such a search?

Of course, yours is far more than a mere circle-packing problem, as you are presumably supposed to minimize cost (which I suppose can be taken to be proportional to the sum of the x-coordinates of the centers, plus the greatest y-coordinate) while also maximizing the total area of the circles, and you have three specified radii available.

The way to start, I think, would be to decide exactly what your goal is (since there are two competing requirements), and how you can efficiently compare two arrangements with respect to that goal; then you can think about how to search for solutions.
I had just asked my Sir on how I should solve this problem:
Firstly he stated, saying I can use any method I deem fit. However, I do not know how to use "Greedy algorithms" let alone computer algorithms in this context.

Furthermore, in respect to you asking about the goal, my teacher replied with saying; start with the greatest area covered and then move onto minimizing piping between each sprinkler. I've already finished with the greatest area covered but the problem is how should I minimize the piping between each sprinkler? I am currently trying to make one goal however it ends up with both goals clashing with one another...

This assignment is here to test whatever method I think I can use to solve this problem. I can use calculus or division; it all boils down too whether or not I got the right answer with well elaborate reasoning for how I go the answer.

#### Karim

##### New member
Since the cost of pipe is given, I am assuming your objective here is to minimize the cost while maximizing the coverage of the sprinklers. When you say "The sprinklers must be connected to a water pipe that runs along the width of the field" does that mean there must be pipe running along two adjacent sides of the field as your diagram seems to indicate?

I personally wouldn't know how to approach this problem other than by trial and error.

I've marked this thread as unsolved.
What I meant by that is I can choose one side of the field where the pipe can run along...It can run along the north side or the south side. From there onwards I extent smaller pipes out of that larger pipe.

#### Karim

##### New member
H
I had just asked my Sir on how I should solve this problem:
Firstly he stated, saying I can use any method I deem fit. However, I do not know how to use "Greedy algorithms" let alone computer algorithms in this context.

Furthermore, in respect to you asking about the goal, my teacher replied with saying; start with the greatest area covered and then move onto minimizing piping between each sprinkler. I've already finished with the greatest area covered but the problem is how should I minimize the piping between each sprinkler? I am currently trying to make one goal however it ends up with both goals clashing with one another...

This assignment is here to test whatever method I think I can use to solve this problem. I can use calculus or division; it all boils down too whether or not I got the right answer with well elaborate reasoning for how I go the answer.
however, I can use analytic geometry techniques to find the length of the pipe required to minimize the cost; but how should I approach it?

#### Karim

##### New member
Since the cost of pipe is given, I am assuming your objective here is to minimize the cost while maximizing the coverage of the sprinklers. When you say "The sprinklers must be connected to a water pipe that runs along the width of the field" does that mean there must be pipe running along two adjacent sides of the field as your diagram seems to indicate?

I personally wouldn't know how to approach this problem other than by trial and error.

I've marked this thread as unsolved.
The diagonal line running across the field gives me a sense of where I should add the sprinklers and it helps me add the sprinklers to the field