Exponents

The Giza Pyramids 3 by Flickr user Tom@HK


As your work receives world attention, people start to realize that you may be right about the usefulness of math.


As your work receives world attention, people start to realize that you may be right about the usefulness of math

You are brought to a briefing in Egypt about a crisis they are having with the Red Sea.


You are told a bacteria that doubles in amount everyday has taken over the Red Sea for the past 5 days.



Apparently the Chinese have developed a pesticide that can kill the bacteria, but they don’t know how much to send.




The Chinese say that it will take 10 days to manufacture the pesticide ( no matter how much is needed, it takes 10 days) and 7 days to ship the pesticide. It takes that long due to the precautions that must be taken during the delivery.




The Chinese say that it will take 10 days to manufacture the pesticide ( no matter how much is needed, it takes 10 days) and 7 days to ship the pesticide. It takes that long due to the precautions that must be taken during the delivery.

However, once the pesticide is there it can be applied all at once and if the right amount of pesticide is there, it should kill all the bacteria within minutes.


You are also informed that each bacteria takes up 0.2m^2 and that the Red Sea is only 438 000 000m^2. You must figure out how much pesticide is needed before the bacteria spreads on to land and the world has a bigger problem on its hands.


So, the question is: How much area does the pesticide the Chinese are sending need to cover if you think it will take you 1 day to make the calculations?

You are also informed that each bacteria takes up 0.2m^2 and that the Red Sea is only 438 000 000m^2. You must figure out how much pesticide is needed before the bacteria spreads on to land and the world has a bigger problem on its hands.

You set to work right away. It’s time to do some serious grunt work, and fast.

You set to work right away. It’s time to do some serious grunt work, and fast.

This is serious, it is real life you’re dealing with now.

It will take 17 days for the pesticides to arrive, it has already been 5 and then the day you are taking to make these equations.


That is a total of 23 days.



First we need to find out how many bacteria there will be by day 23. Then we need to find out how much area that will cover.




It will take a long time to calculate all this by doing:

1 * 2=2

2+1=3

3 * 2=6 etc.




It will take a long time to calculate all this by doing:

1 * 2=2

2+1=3

3 * 2=6 etc.

So, make a grid where each day represents one square on the grid ( sort of like a calendar), and applying the same concept as Pascall’s Triangle, see if you can identify any helpful patterns within this grid.

BestChess.com.au, chess board- Google images

If you think about this question, you may notice that it is similar to the “rice on the chessboard” problem.



A famous math problem where the man who invented chess, as a reward requests that he receives a grain of rice for every square of the chessboard. However, he also says that he wants, for every square, double the amount of rice on the previous square.



A famous math problem where the man who invented chess, as a reward requests that he receives a grain of rice for every square of the chessboard. However, he also says that he wants, for every square, double the amount of rice on the previous square.

So, the first day he has one grain of rice. The second 3, the third 7 etc. Well, eventually the kingdom runs out of rice because there is just not that many grains of rice in the world. By the 64th square the man would have 1.8 * 10^19 grains of rice, so his request is never fulfilled.

So, to build your chart, build it like a chessboard. It is easier to see and you will be able to refer to the rice on the chessboard problem.

So, to build your chart, build it like a chessboard. It is easier to see and you will be able to refer to the rice on the chessboard problem.

Remember that each square goes up by a multiple of 2. The first square starts off as 2^0 so the last square should be 2^63.

2^0

2^1

2^2

2^3

2^4

2^5

2^6

2^7

2^8

2^9

2^10

2^11

2^12

2^13

2^14

2^15

2^23

2^22

2^21

2^20

2^19

2^18

2^17

2^16

2^24

2^25

2^26

2^27

2^28

2^29

2^30

2^31

2^39

2^38

2^37

2^36

2^35

2^34

2^33

2^32

2^40

2^41

2^42

2^43

2^44

2^45

2^46

2^47

2^55

2^54

2^53

2^52

2^51

2^50

2^49

2^48

2^56

2^57

2^58

2^59

2^60

2^61

2^62

2^63

You can see here that 2 to the power of the day you’re looking for minus 1 gives you how much more bacteria was made that day. However, this does not tell you how much bacteria you have in total up to that particular day. So let’s make a T-chart.

So, from this T chart we can see some patterns. The interval between the number of bacteria each day is increasing by a multiple of 2.

1+2=3

3+4=7

7+8=15 etc.

So, from this T chart we can see some patterns. The interval between the number of bacteria each day is increasing by a multiple of 2.

1+2=3

3+4=7

7+8=15 etc.

We can derive and equation from this.

If you take the day you’re on and subtract 1 and raise 2 to that power you will have the amount of bacteria produced on that one day.


Now take this value and subtract 1 from it to get a second value. Take this second value and add it to your first and that will give you the TOTAL AMOUNT of bacteria up to the day you’re on.



This equation would look like this:



(2^(D-1)) + (2^(D-1)-1)= Total Amount of Bacteria

This equation would look like this:

Where D is the day you’re on.


So, we want to find the amount of bacteria on the 23rd day. Well, 23 is our D so lets plug it into the equation.



(2^(23-1)) + (2^(23-1)-1)= Total Amount of Bacteria

2^22 + ((2^22)-1)= 8 338 607 Bacteria




2^22 + ((2^22)-1)= 8 338 607 Bacteria

Now, we know each piece of bacteria takes up 0.2m^2. So, if we multiply this by the amount of bacteria we have we will get the total area the bacteria takes up.




2^22 + ((2^22)-1)= 8 338 607 Bacteria


0.2m^2(8 388 607)= 1 677 721.4m^2




2^22 + ((2^22)-1)= 8 338 607 Bacteria


0.2m^2(8 388 607)= 1 677 721.4m^2

So, the Chinese need to send enough pesticide to cover 1 677 721.4m^2. Lets hope it will work and your calculations are correct, or we’re all in trouble.

17 days later....

17 days later....

The World is saved because of you and your math. The bacteria is all destroyed and the world does not have a crisis on its hands. The people have restored their faith in math.

Exponents

Education

Transcript of Exponents