Question 1169917: Compare the doubling times found with the approximate and exact doubling time formulas. Then use the exact doubling time formula to answer the given question.
A family of 200 termites invades your house, and its population increases at a rate of 21% per week. How many termites will be in your house after 1 year (52 weeks)?
Calculate the doubling times found with the approximate and exact doubling time.
The approximate doubling time is "blank" weeks and the exact doubling time is
"blank"weeks?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! from what i've read, the approximte doubling growth rate is found by dividing 70 by the percent per time period growth percent.
in this problem, time periods are in weeks, and the growth rate percent per week is 21%.
applying the rule of 70, the population will double in 70/21 = 3.3333... weeks.
there are 52 weeks in the year.
to find the number of doubling time periods, divide 52 by 3.333... to get:
number of doubling time periods in 52 weeks is 15.6 times.
if you start with 200 termites, they will double 15.6 times in 52 weeks.
the formula for that is f = 200 * 2 ^ 15.6.
this results in f = 9933400.09 termites in week 52.
the exact doubling formula is ln(2) / ln(growth rate) = ln(2) / ln(1.21) = 3.636270449.
the number of doubling periods in 52 weeks is equal to 52 / 3.636270449 = 14.30036647
if you start with 200 termites, they will double 14.30036647 times in 52 weeks.
the formula for that is f = 200 * 2 ^ 14.30036647.
this results in f = 4035238.905 termites in week 52.
the approximate formula gave you 9933400.09 termites in week 52.
the exact formula gave you 4035238.905 termites in week 52.
if you did not use the doubling formula, but used the weekly growth rate formula directly, then you would have gotten:
f = 200 * 1.21 ^ 52
this results in f = 4035238.905.
this is the same as you got with the exact doubling time formula, which is no surprise, since the exactly doubling formula is determined from the weekly growth factor.
to summarize:
the approximate doubling factor was 70/21 = 3.333...... weeks.
the exact doubling factor was ln(2) / ln(1.21) = 3.636270449 weeks.
using the approximate doubling factor, the number of termites in 52 weeks is equal to 200 * 2 ^ (52/3.333.....) = 200 * 2 ^ 15.6 = 9933400.09
using the exact doubling factor, the number of termites in 52 weeks is equal to 200 * 2 ^ (52/3.63627049 = 200 * 2 ^ 14.30036647 = 4035238.905.
using the weekly growth factor,the number of termites in 52 weeks is equal to 200 * 1.21 ^ 52 = 4035238.905.
i'll be available to answer any questions you might have about this.
theo
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