** **

# Barak wanted to join a baseball team

Barak was as likable a boy as anyone else. In fact a neighborhood girl named Ayesha liked him a lot and she was always with him. Barak was what you might call – a little obese. Some of the children nicknamed him fatso. He was a good natured boy and accepted the tease graciously although Ayesha dealt sternly with the person who teased him in her presence. Ayesha did not say she was his girlfriend. Nevertheless, she always wanted to see Barak happy and liked him dearly.

Barak had a baseball bat and played the game in the street with his friends. When Barak turned 12, he wanted to join the junior league team. He went and registered for it. The coach told him that there are 20 people interested in joining the team but he could take only 15. He had to do this selection to make sure that every player in the team got a chance to play. The coach thought that a fair way is to select players by their running time on a 1 kilometer (km) track. That showed both their speed and stamina. His logic was that in order to be a good fielder you had to be able to run at a good speed many times a day. He was prepared to work with the players to develop their other skills – catching, throwing and batting. When the 20 children ran the 1 km track, Barak was dead last, You could say that he hardly ran because it took him 15 minutes to complete for the 1 km run. Some people even walk much faster than that.

# Barak ran too slow to make the team

Barak went home with a long face because he failed to make the team. Ayesha saw him.

Ayesha: Barak, why the long face? Did you register for the junior league?

Barak: No, the coach said that I am too slow. I can try it again next year.

Ayesha: What did exactly happen?

Barak: It is embarrassing. I completed my 1 km track in 15 minutes. I was last. Everybody ran past me. Some people finished the run in 4 minutes. I talked to the coach afterwards. He said that if I can run 1 km in 6 minutes, I will have a good shot at it next year. How am I going to go from 15 minutes to 6 minutes?

# Ayesha agreed to coach Barak in running

Ayesha: I will help you because I have faith in you. I am sure, you will make it.

Barak: Thanks, but how useful will be your faith? Do you have a plan in mind?

Ayesha: I will become your running coach and if you want I will also be your running partner. I hope you like chasing after me (giggling). You already did the run today. Let’s start tomorrow evening.

Next day evening, Ayesha went to Barak’s house to make sure that he ran.

Ayesha (on the way to the running field): I know you completed 1 km yesterday in 15 minutes which is 900 seconds. You run most of it as you did yesterday but the last 100 meters, I want you to chase me.

So they did as per Ayesha’s plan and Barak ended up completing the track in 3 seconds less than the previous day. Ayesha made him repeat this plan the next day but this time he had to chase her for the last 200 meters. They continued this process for 1 week and Barak’s timing came down from 900 seconds to 882 seconds.

# Ayesha proposed exponential decrease in running time

Ayesha: Barak, see I have faith in you. You brought your timing down by 2% in one week. Your time is 0.98 multiplied by the time a week ago.

Barak: Yes, but where does that get me?

Ayesha: Next week, you will bring your timing to 0.98 times of what it is today.

Ayesha and Barak went and practiced every day. Surely, Barak succeeded in this goal. His timing was now 864 seconds which was 0.98 multiplied by 882 seconds. This went on for 4 weeks. Barak’s timing was now down to 830 seconds.

Ayesha: Barak, by bringing your timing to 0.98 times that of the previous week consistently, now after 4 weeks your timing was 0.922 times of your original timing.

Barak: Yes but it is still a long way from 6 minutes. I don’t know why you are so sure.

Ayesha (as if teasing); When you chase girls they don’t slow down for you, I want to make sure that you are good and ready to chase me when the time comes. I am joking. I have to tell you something first though.

Barak: What is that?

# Math teacher said it should work

Ayesha: I talked to our math teacher about this.

Barak: It would have made more sense if you talked to our physical education teacher.

Ayesha: No, the math teacher. The plan I set up for you is a decreasing exponential function. In this case it is an exponential time series with the base 0.98, the value of the series will be: Initial value x 0.98^{number of weeks}. The initial value was 900 seconds. Now for 4 weeks the expected value would be 900 x 0.98^{4 }which comes out to 830 seconds which is your timing today. So you showed me that it works. Because, this value in 4 weeks is 0.922 times the original value, I can write this also as: Initial value x 0.922^{number of 4 week periods}. In the next 4 weeks, I expect it to go to 900 x 0.922^{2} which will be 765 seconds. If you do that, it will show that our plan is definitely working.

Barak: It is still far from the 6 minutes timing I need.

Ayesha: If we keep doing this, in 52 weeks your timing will be 315 seconds which is way better than the 360 seconds (6 minutes) that you need. The math teacher even made this graph for me (see picture of Barak’s Training). Of course, I assume that you will keep working with me.

Barak: Thanks for your help Ayesha. I will keep working with you all my life if you get me in the junior cricket league next year.

Barak wanted to grab Ayesha and hug her but he did not. He thought that Ayesha may quit if she takes it the wrong way.

* BODMAS or PEMDAS*

Ayesha calculated the power function first and then carried out the multiplication. For example for 900 x 0.922^{2}, she first calculated 0.922^{2 }to be 0.85 and then multiplied it by 900 to get 765. Also in the previous stories, multiplication or division was done first and then the addition or subtraction. In general the order in which these calculations are done is called BODMAS (brackets, order (power), division/multiplication, addition/subtraction) or PEMDAS (parentheses (brackets), exponents, multiplication/division and addition/subtraction).

*Challenge*

Barak does well and decreases his 1 km run time each week to 0.98 times that of the previous week. He continues this for 52 weeks (see the graph above), and is very proud of it. Now, he wants to continue this running training with the same condition for an additional 52 weeks so that his total training duration will become 104 weeks. Draw a graph of his run time over the 104 week period. To make life easier, use the conversions 0.98^{13 }= 0.769 and 104 weeks = 8 thirteen week periods. Do you think he will achieve this goal? Hint: check the world records.

*Solution:* The values for 0,13,26,39,52,65,78,91,104 weeks are 900, 692, 532,409,315,242,186,143,110 seconds.

Graph for run time with the function

Run time = Initial run time x 0.98^{weeks}

Calculated every 13 weeks for a total of 104 weeks.

Note that the predicted run time is 110 seconds. The world record for 1 km run set by Noah Ngeny in 1999 is 131.96 seconds.