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 = ae^kt

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 = ae^kt

y/a = e^kt

Take the logarithms of both sides of the equation

log(y/a) = log(e^kt)

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