Ashley and Brittany – the cheerleaders
Ashley and Brittany were close friends – well, as close as two swaggering teenagers can be. Both were extremely popular in school. They were cheerleaders. Of course, their athletic bodies, expensive haircuts and fashion forward clothes added to their popularity. They appeared to respect each other. One would guess that they had to because each of them knew so many secrets of the other.
Ashley had a major weakness – a sweet tooth. She would consume as much candy as she could lay her hands on. One day, Ashley and Brittany were just sitting and talking to each other.
Ashley: I gained 5 pounds last month. I eat healthy, do cheerleader practice and work out with you at the gym. I don’t want to gain any more weight and become a whale. I think some of the boys are losing interest in me already. Josh has even started to pooh-pooh me.
Brittany: Do you really think that you eat healthy?
Ashley: Are you talking about the candy? I have had a sweet tooth ever since I was little. I can’t help it.
Who could be more popular than us?
Ashley started brooding and cursing relentlessly to the only thing that was sweet in her that is her sweet tooth. Brittany showed Ashley a group of students. They were all crowded around and talking to this one girl. Who could be more popular than us, they both wondered jealously. They decided to find out. They called one of the boys in the group and asked him about this girl. He told them that the girl’s name was Sara. He added that Sara was very smart and nice to everyone. Brittany and Ashley could not believe that this girl could be so popular. She was not even a cheerleader, and she was dressed only like an ordinary girl – no expensive haircut, no high heels, nothing. They decided to check out why the girl was so popular. Was she really that smart? They went to the group of those students to talk to Sara. Of course, Sara’s boyfriend Johnny was there too.
Ashley: Are you Sara? I have heard a lot about you.
Sara: Yes, my name is Sara. This is my boyfriend Johnny. What have you heard about me?
Ashley: I am Ashley. This is my friend Brittany. We were just talking about you and how popular you are.
Sara: Thanks. I didn’t know that I was popular.
Brittany: Popular and smart too.
Sara: Please, stop embarrassing me.
Ashley tells Sara about her sweet tooth
Ashley: I have a problem and I thought that being a smart girl you might be able to help me.
Sara: What kind of problem? I will try to help if I can, but no promises.
Ashley: I gained 5 pounds last month. I eat healthy, do cheerleader practice and also regularly go to the gym with Brittany. I don’t want to keep gaining weight. The problem is that I have had a sweet tooth ever since I was little. I can’t help it.
Brittany: You can’t quit suddenly. How about if you did it slowly?
Sara’s plan for Ashley
Sara: Brittany is right. I have a plan for you. Make a promise to yourself that on any one day, you will eat only half of the candy you have on you, and that you will not buy any more candy.
Ashley: Huh, that’s smart. That way I would quit slowly. Do you think it will work?
Brittany: It’s worth a try. If it works, you will again become the dream girl of all the boys. If it doesn’t work, we will tell everyone that Sara is not that smart.
This was enough incentive for Ashley – becoming the dream girl. She could even overlook that this would prove Sara to be very smart. She went home, took the kitchen balance and weighed all of her candy. It was 256 g. That day she ate 128 g of the candy which was half of the 256 she had with her. Actually, that was almost the same amount of candy that she was used to eating every day, anyway. Now, there was 128 g of candy left. The next day she ate half the amount, and there was 64 g left. She kept eating half of the remaining candy each day until only 1 g was left.
Ashley was happy, boasting to Brittany as to how she had used her determination, and that only 1 g of candy was left. She was thinking of boys like Josh coming back to admire her beauty.
The Valentine’s Day
There was one thing that Ashley had not counted on -Valentine’s Day. She still had many friends. Some of her friends were anxious to stay on Ashley’s good side and knew how much she loved candy. None of them were considerate enough to know about Ashley’s sweet tooth as a problem and about her current plan. They brought her chocolate and other candy. All the candy was wrapped in papers with heart stickers on them. How could she turn them down? All these friends loved her.
Brittany: Ashley, you are very popular.
Ashley: What am I going to do about the candy and my sweet tooth?
Brittany: Give the candy to your kid brother.
Ashley: No way, how can I give the candy to that punk? The cute boys gave it to me with love. I can almost taste their kisses in the candy.
Brittany: You must have a kilogram of candy. You are not going to stick to your old rule of eating half of the remaining candy each day. Are you?
Ashley went back to Sara and said: I followed your plan of eating only half of the remaining candy every day. It was working, and I was almost done but then the Valentine’s Day came and I got a lot of candy from the boys who love me. Now, I am stuck.
Sara looked at the amount of Candy Ashley had, smiled and said: I never thought that being so popular could be a curse. For sure, I have the solution but I have a class right now. When can we meet?
Ashley: Are you free at 2? Maybe we can meet in the cafeteria.
Sara: Yes. See you at 2 but don’t eat any candy until then.
Was Sara smart enough to solve this problem?
Ashley and Brittany did not believe that Sara was smart enough to do anything to solve this problem. Nevertheless, they decided to meet her. Maybe they could show her up if she did not have a good solution. They met in the cafeteria. Sara had her laptop with her.
Sara: Let me start by showing you a graph for what you were doing before the Valentine’s Day. This type of curve is called an exponential decay. This decay has a half life of 1 day because you had decided to eat only half of the remaining candy on each day.
Brittany: I get that. Don’t give me any of the math jumbo-mumbo. How does it help the current situation when Ashley has almost 1 kg of candy?
Sara: Simple, we keep the same plan but we will change the half life from 1 day to 1 week.
Brittany: Are you telling me that she will eat half of all the candy in one week, and then half of the leftover next week? That’s smart. The plan might work.
Ashley had weighed all the candy including the 1 g she had left from before. The total weight was 1024 g.
Ashley: Thanks for the new scheme. I keep the same plan but just change the time of 1 day to 1 week.
Brittany: How will you decide how much candy to eat today ? Can you eat the whole week’s worth of candy on one day, and then wait for the next week to come?
Ashley: That would be both stupid and hard to do. Sara, what do you think?
Sara: One week has 7 days. Sara: Here is the amount of candy you should eat from days 1 to 7: 97g, 87g, 79g, 72g, 65 g, 59 g and 53 g. This is a total of 512 g out of the 1024 g you have. So you will have 512 g left. I will make you a chart for all the different days after the first week.
Ashley: I could do that. Thanks Sara. Now, I know why everyone likes you.
Brittany: Some time you should come and watch us do our cheers. This is an invitation from both of us.
Sara: Thanks and I accept the invitation happily. Johnny and I will come together to watch you guys.
All along Johnny quietly watched what Sara had done. After school, Sara and Johnny talked about it.
Another application of exponential functions
Johnny: I see that you found one more application of the exponential functions.
Sara: Only one thing is different, last time we talked about exponential functions as am when “a” was greater than one. Nana used to get double the amount of money each time, So we used the function f(t) = Cinitial x am with Cinitial being the amount she got on her birth which was one dollar, a = 2 because the amount was doubled each year and m being the number of years. I told Ashley that she could eat half the remaining candy on each day. Therefore, here a =1/2.
Johnny: I know the amount of candy she can eat on any day will become
f(t) = Cinitial x am = Cinitial x (1/2)(number of days).
Sara: Johnny, you remember we can also write 1/2 as 2-1. Therefore,
f(t) = Cinitial x (2-1)(number of days) which is the same as Cinitial x 2-number of days.
Johnny: I get it but we can also write in a general form as:
f(t) = Cinitial x 2(-time/half life).
Sara: That’s what I did when I told her that she could eat half the remaining candy every week except that I changed the time to weeks instead of days. Then it became:
f(t) = Cinitial x 2(-weeks/half life in weeks) and with half life being 1 week, I got
f(t) = Cinitial x 2(- number of weeks) = Cinitial x 2(-7 x number of days). That’s how I got the numbers I gave her after the Valentine’s day.
C(t) = 2(-time in days / 7).
I see that the second exponent decays much slower than the first one (Fig. 5.1). That makes sense. I suppose you will round off the numbers to the nearest gram when you tell her the amounts to eat on the remainder of the days.
Johnny was impressed with Sara’s ability to find this new use for the exponential functions. He could not resist making a crack, “I suppose exponents are like mountains. With a positive power to you can reach higher heights but you slide down when the power is negative.”
Sara: That’s it but remember it works only for a base greater than 1. For an exponent with a base less than, you slide down the mountain.
Tony Gordito, a grade 10 student, was sitting in the school cafeteria with his friends and enjoying the food. Chelsea said that she was going to run in the 5 km race on that weekend to raise money for the Cancer Foundation. Tony said that maybe he should run too. Everybody laughed thinking that Tony could not run. Chelsea asked, “How fast?” Tony countered, “How many minutes do you take for the 5 km run?” Chelsea said, “22 minutes”. Tony declared that he would run next year and beat her 22 minute time. Now that he had made the declaration, “Tony went to the tracks to start the training.” He walked as fast as he could, and with occasional jogs in between. he completed the 5 km in 60 minutes. He figured that he could practice 2-3 days a week and every week he would take 2 percent less time than the previous week. If Tony remains determined to do this, do you think that after 52 weeks, he could meet the challenge of running 5 km in less than Chelsea’s 22 min?
Solution: Tony ran the 5 km in 60 min on the weekend. After each week he will take 2% less time than the previous week. Therefore the time after n weeks will be
60 x (1- 0.02)n or 60 x 0.98n.
So after 52 weeks his timing will be 60 x 0.9852 minutes which is 20.985 minutes. That gives him a little more than one minute under the challenge of running in Chelsea’s time of 22 minutes.