Exponential growth problem - explained two ways
print resourcePosted by: Laura Avram Partner: Seeds of Hope
Theme: Math;teaching Age Group: 6-12 Date Added: October 15, 2011
Summary
This is an exponential growth problem, explained in both an intuitive and mathy way.
Objectives
Students learn about exponential growth
Main Content
EXPONENTIAL GROWTH
The problem : A plant growing on a lake doubles its surface every day. At the end of the month it is going to cover the whole surface of the lake. How many days is it going to take two plants to cover the whole surface of the lake ¿
a) One day less than a month
b) Two days less than a month
c) Half a month
d) One day more than half a month
METHOD A – intuitive method
If it takes a plant a day to double its surface, that means we are “one day ahead” when we start directly with 2 plants. In a hypothetical first case, in which we start with one plant, we are going to have two plants by the second day. However, in a hypothetical second case, we start directly with two plants, which means that in the second situation we are one day ahead, thus will get to the end (the whole lake covered) one day earlier. Since in the first situation, it takes one plant a month to cover the whole surface of the lake, it will take us one day less than a month when we have two plants.
Answer : one day less than a month
METHOD B – mathy method
Situation I : starting with one plant and assuming a month has k days
DAY 1 – 1 plant (20 plants)
DAY 2 – double the amount from DAY 1 – 2 plants (21 plants)
DAY 3 – double the amount from DAY 2 – 4 plants (22 plants)
…
…
We notice that there is a relationship between the number of the day and the power of 2 – the power of the 2 is the number of the day minus 1 . 0 = 1-1, 1 = 2-1, 2 = 3-1
Thus,
DAY K – 2 K-1 plants
Thus the surface of the lake is 2 k-1. Now we need to see how many days it will take us to get to the same surface when we start with 2 plants (it took us K days in this situation.)
Situation II : starting with two plants and assuming a month has k days
DAY 1 –2 plants (21 plants)
DAY 2 – double the amount from DAY 1 – 4 plants (22 plants)
DAY 3 – double the amount from DAY 2 – 8 plants (23 plants)
…
…
We notice that there is a relationship between the number of the day and the power of 2 – the power of the 2 is the same as the number of the day.
Thus we are going to reach 2k-1 after the same number of the days as the power of two : k -1 . Since it took us k days (the number of days of a month) to cover the whole surface in the first situation, that means that in the second situation (starting with 2 plants as opposed to starting with one) it takes us k – 1 days, which is one day less than a month.
Answer : one day less than a month