Question 112151: This question deals with an exact formula. In the following exercise use the approximate and exact formulas to find the doubling time. Also answer the given question.
38. 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?
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%. 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.
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 38. 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?
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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
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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
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Cheers,
Stan H.
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