Priya the go getter
Priya had always been a go-getter. Now, at the age of 16, she worked with her mom Jenna in her shop that printed and sold T-shirts with designs printed on them. They made T-shirts for anybody who came there, and also took special orders from teams. Priya made slightly more than the minimum wage and worked only part time at the store because she was still going to high school.
Priya was taking a Geography course. The final exam was worth 40% of the course mark. The teacher offered everyone in the class the option of doing a Geography project in lieu of the final exam. Priya had very low marks going into this exam. She thought that she would be better off doing the project on which she could work at her own pace.
Priya’s clever geography project
Priya was a very clever girl. She proposed to the teacher that she would do a project on: “from where the customers come to her mom’s T-shirt shop “. She thought that she could do this project at work on Jenna’s time. Hoping that it might help improve the business, her mom agreed to it. The Geography teacher approved the project considering that it came close enough to being a Geography assignment, and also because the idea was a practical one.
Customers gave their addresses and phone numbers whenever they ordered T-shirts from the shop. They would come and pick up their orders on or after the due date. Priya took out the shop records from the last two months. For each customer, she looked up the address and found the distance between that address and the shop using “Google map” on her lap top. She recorded these distances as her data. She divided the data as follows: customers who came from less than 1 kilometer away, between 1-2 kilometers away and greater than 2 kilometers away. She concluded that very few customers came from more than 2 kilometers. She wrote her project report with this conclusion.
The teacher said, “You can do better, much better”
She asked her Geography teacher to see the report and give his opinion without actually marking it. The teacher glanced through the report and said, “You can do better, much better”. The teacher also said that because this was an independent project, she should not ask him for help.
Poor Priya was sitting in the school cafeteria with her head down. She was perplexed. She didn’t know what to do. Her friend Kate came but Priya was in no mood to chat. Kate was a good friend and didn’t give up on her, and sat down anyway. Kate asked Priya why she was in such an awful mood.
Priya: It’s my Geography project. The teacher is not happy with it. I might fail Geography.
Kate: Let me see the report.
Priya gave the report to Kate who took a quick look but did not know how the report could be improved. Priya thought that she was doomed. Suddenly, Kate started to speak.
Kate introduced Priya to Sara
Kate: There is Sara. Why don’t you talk to her? May be she can help you. She is a real whiz.
Priya: For one thing, I don’t know if she took Geography or not. Second, why would she help me even if she could? She is two years senior to us and doesn’t even know me.
Being a good friend Kate did not quit. Instead she was rather pushy said: She’s really smart. She loves a challenge. Besides, what do you have to lose by talking to her. Be nice to her. I talked to her one time. I will introduce you two.
Kate went where Sara was sitting, said hello to her and sat down. Priya followed Kate.
Kate: Hi Sara. I am Kate. We met once but I don’t know if you remember me. This is my friend Priya.
Sara; Hi Kate, I remember you. Nice to meet you Priya.
Priya: I have heard a lot of good things about you. Everyone at school knows how you helped Johnny with Trig.
Sara: Yes I did. He’s my boyfriend. Is that a problem?
Priya: I do have a problem but it is not that you are smart and that you helped your boyfriend. I admire you for it. I wish I had someone smart as a friend who could help me.
Sara: You said you had a problem.
Kate: I think she has a challenge for you.
Priya: Look, I do have a problem and I came to you because you are smart. I will understand if you think I shouldn’t have come. You must have a lot of work to do.
Sara: First, let me hear your problem, and then I will decide if I can help you or not.
Priya told Sara about her Geography project and showed her the report. Sara quickly went through it.
Sara: I know how this project could be improved.
You said this is for a Geography course. Then put some Geography in it. You said that you had addresses of the customers. Take an area map. Mark your shop on it. Then mark each customer on the map with a dot.
Priya: Thanks, I could easily do that.
Sara: Where is your shop?
Priya gave her the exact address.
Sara smiled and said: Do you want an A+ on the project?
Priya: Yes, how?
Sara: First, bring the map with all the dots and then I will tell you.
Priya: Thanks Sara. I will do that.
Priya was on cloud 9 with the mention of getting an A+ and thanked her best friend Kate for introducing her to Sara. She worked hard for three days and put a dot for the address of each customer on the area map, and a square for the shop location. Then she brought the marked map to Sara (Fig.8.1).
Sara: Doesn’t it look more like a Geography project now? You have a map with a river and everything. Is your mom happy with it?
Priya: My mom said that this map told her a ton more than the last report did. We hardly have any customers from the other side of the river.
Sara: The river was the reason why I smiled the last time when you told me the address of your shop. If you write a report based on this map, your teacher should easily give you a B+ plus or even an A– but to get an A+ you will have to learn something new. I have been thinking about it since we talked last time.
Priya: What do I have to learn?
Sara: I can work with you but you have to pay attention even when it gets hard.
Priya: I am all for it. Bring it on.
Sara turned her laptop on, and brought a picture of a dart board.
Priya: Do I have to learn how to play darts? I already know that.
Sara: No, we are going to use the dart board just to learn something.
Priya: Okay, shall we start with the bull’s eye? That’s hard to hit with a dart.
Sara: Remember, in Math you learned about X-axis and Y-axis? The origin was the point where the two axes crossed.
Priya: Yes, I remember that. Are we going to call the bull’s eye the origin?
Sara: Yes, we will say that the inner bull’s eye is the origin. There are many circles around it. Which circle is closest to the origin?
Priya: The outer bull’s eye is the closest circle.
Sara: Can we say that the outer bull’s eye is a circle with the smallest radius?
Priya: Yes. The circle with the next larger radius is the triple ring. The largest radius is for the double ring.
Sara: That’s great. Then we can define the positions of the rings as their radii with the inner bull’s eye as the origin.
Priya: Okay but I thought I just did that.
Sara: Yes but now I am going to show you the dart board with a dart on it. I took out the scoring numbers from this dart board. Now, tell me the position of the pin of the dart.
Priya: It’s in the outer ring directly above the inner bull’s eye.
Sara: Exactly. Now you told me the position of the dart. If we can write East, West, North and South, you can tell me the position of the dart in a more precise way.
Priya: Yes, now I can say that the pin of the dart is North of the inner bull’s eye.
Sara: If I put the dart some place which is not exactly in any of the directions written there but at a different angle, it will be harder for you tell me the position of the dart.
Priya: Can’t we write positions of different angles to make that easy?
A graph paper with polar co-ordinates
Sara: Yes, we can. You have just asked me for a graph paper with polar co-ordinates.
Priya: Did I? I don’t even know what they are.
Sara: Polar co-ordinates tell you the distance from the origin and the angle where a point is. The angle starts at East as 0° and then goes to North as 90°, West as 180° and South as 270°.
Priya: That’s neat. If the dart was anywhere on this graph, I could have said exactly where it was – how far from the origin and at what angle.
Sara: The co-ordinates for the dart would be written as: r, θ where r is the distance from the origin and θ is the angular location. In books, you will also see the word modulus for r and argument for θ.
Priya: What is the big deal of the polar co-ordinates over the regular x, y (Cartesian) co-ordinates we learned in geometry? Can they be converted to each other?
Sara: They are both used for different purposes. Polar co-ordinates are used more in geography because the earth is somewhat like a sphere. They are also used in navigation, for radars and anywhere else one wants to understand curves. Yes, you can convert Cartesian co-ordinates into polar and also polar co-ordinates to Cartesian using Trigonometry.
Priya’s map in polar co-ordinates
For now, let’s work on improving your project score. I want you to overlay the polar co-ordinates on the map that you showed me earlier. Put your shop at the origin.
Priya: From the picture in polar co-ordinates I see that most of our customers come from θ = 90° to 360° which is marked here as 0°. There are very few customers from θ = 0° to 90°. In θ = 90° to 360°, more customers come from nearby and fewer from far off. I don’t know how to say this part better.
Sara: For the second part you can say, “There is an inverse relationship between distance and the number of customers.” Now we have to say why there are very few customers from θ = 0° to 90°. This is because there is a river running very close to your store at that place. What location do you think would be the best for your shop if you could move it?
Recommendations based on the study
Priya: A shop close to the bridge but still on the same side will be the best location because then our current customers will be still close by and keep coming and we will also get some more customers from the other side of the river. I think if possible and if we can afford it, that location would be great.
Sara: Now, write all the observations and recommendations in your report. Also, what is the title for your report?
Priya thought for a while and then said: “The Effect of Geographical Factors On Success Of A Business.”
Priya rewrote the report based on what she had discussed with Sara. She submitted the report to the Geography teacher and also gave a copy to her mom.
Priya got 100% on the report. She was very happy. This was the first time ever she got 100% on any assignment or test. Mom was also glad to see the great job Priya had done. She had given her an actual suggestion for improving the business.
Priya came to visit Sara and gave her the thrilling news that she got 100% on the project. She thanked Sara for all her help. As a token of appreciation, her mom told her to give Sara two T-shirt coupons: maybe one for Sara and the other for her boyfriend.
Sara: Thanks but you didn’t have to do that. I loved working with you. That’s how I got to learn about polar co-ordinates myself. Priya said bye to her new friend and left.
Challenge: Priya told her 14 year old brother Joga how she used polar coordinates to get 100 percent in her geography assignment. Joga, told Priya, “Thanks for the idea, I could use it.” Joga was supposed to mow the lawn. So he went to the yard. In the middle of the yard, he hammered into the ground a large peg of wood with a radius of 7 centimeters. He stretched the other end of string all the way, and tied to it a lawn mower which was now 8 meters away from the center of the peg. After he started the mower on an autopilot, it started moving around the peg. The string was now winding around the peg. Then he called Priya and said, “See polar co-ordinates. Now, you draw the path of the lawn mower.” What do you think Priya’s sketch looked like?
Solution: The peg has a radius (r) of 7 cm. The circumference of a circle is 2πr or approximately 2 x 22/7. The circumference of this peg is about 7 x 2 x 22/7 or 44 cm. At start, the lawnmower is 8 meters away from the peg. As it mows, it cannot go away, it tries to make a circle but as it goes around, the string winds around the peg up at the rate of 44 cm per revolution. The path will become shorter by 2πr with the movement of an angle of 2π radians. If the value of the angle θ is in radians, the path will be a spiral with the equation
r = 800 – 7 θ cm. Here is the sketch of the path from 0 to 30 π radians which is 15 rounds of the spiral.