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Homework answers / question archive / If the number of telephone solicitations increases exponentially (continuously), how long will it take to double your number of annoying phone calls if the rate of growth is 12% per day?
If the number of telephone solicitations increases exponentially (continuously), how long will it take to double your number of annoying phone calls if the rate of growth is 12% per day?
5.7762 days
Step-by-step explanation
The general formula of continuous exponential growth and decay is:
y = aekt
where:
y = final amount
a = original amount
e = Euler's number ≈ 2.71828
k = growth rate constant
t = time
Deriving a formula from this general formula to calculate for the time t:
y = aekt
y/a = ekt
Take the logarithms of both sides of the equation
log(y/a) = log(ekt)
Take down exponent kt as a coefficient
log(y/a) = (kt)log(e)
t = log(y/a)/klog(e)--> formula to solve for time t where y/a is the ratio of the final amount to the original amount
In the problem,
y/a = 2 (the ratio is 2 since the amount doubled)
k = 12% = 0.12
e = 2.71828
Using our derived formula:
t = log(y/a)/klog(e)
t = log(2)/(0.12)log(2.71828)
t = 5.7762 days