A family of 100 termites invades your house and grows at a rate of 20% per week.

Question

A family of 100 termites invades your house and grows at a rate of 20% per week. How many termites will be in your house after 1 year?

a)Use the rate formula to find the number of termites after 1 year.

b)Use the rate to find the approximate Tdouble, and then find the number of termites in 1 year, using the Tdouble – new value formula.

c)Use the rate to find the exact Tdouble, and then find the number of termites in 1 year, using the Tdouble- new value formula.

Compare the values from a), b), and c.

I have received 2 previous answers that have been marked wrong by my professor. Please help?

Explanation / Answer

a)

So, 100(1+20%)^52 = 100(1.2)^52 = 1,310,463 termites at the end of the year and a holey house.

b)

The approximate doubling time formula (rule of 70): for a quantity growing exponentially at a rate of P% per time period, the doubling time is approximately Tdouble = 70/P. This approximation works best for small growth rates and breaks down for growth rates over about 15%.
T(double) = 70/20= 3.5 per week

T(double) = 3.5*52 = 182 per year

Number after a year = 100 * 2^(52/3.5) = 2967875.303

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c) This is the exact doubling time formula: For an exponentially growing quantity with a fractional growth rate r, the doubling time is Tdouble = log10 2/log10(1 + r). For an exponentially decaying quantity, we use a negative value for r (for example, if the decay rate is 5% per year, we set r = -0.05 per year); the half-life is Thalf= - log10 2/log10(1 + r).

Note that the units of time used for T and r must be the same. For example, if the fractional growth rate is 0.05 per month, then the doubling time will also be measured in months.
T(double)= log(base 10)2/log(base 10)(1+e)
T(double) = 0.3010299957/log(1+0.20) = 3.801078... per week

T(double) = 3.801078...*52 = 197.6927.. per year

Number after a year = 100 * 2^(52/3.8) = 1316308.954

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