SOLUTION: 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 fami

Algebra ->  Exponents -> SOLUTION: 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 fami      Log On


   



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) About Me  (Show Source):
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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.