Question

a growth rate parameter of per hour. How many hours does it take for the size of the sample to double

Answer 1

The equation for exponential growth in terms of a percentage growth rate is:

x(t) = x(0) * (1 + r/100)^t

where x(t) is the value of the exponentially growing quantity at time t
x(0) is the initial value of x at t = 0
r is the percentage growth rate per unit time
t is time, expressed in the same units as used to express the percentage growth rate.

The doubling time is the time when x(t)= 2*x(0). Plugging this into the above equation, along with the growth rate you were given yields:

2*x(0) = x(0) * (1.04)^t

2 = 1.04^t

Take the logarithm of both sides to get:

log(2) = t*log(1.04)

t = (log(2))/(log(1.04))

(it doesn't matter what base you use to take the logarithms)

t = 17.67 hrs
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Mr. Algebra's answer is pretty much nonsense.

Ivan T. had the right idea, but was off by one increment in time. After 1 hour, there is 1.04 times the original amount of sample present. For some reason, he thinks this is the amount present after 2 hours.

Pointy didn't recognize that the growth rate in this question is given as a percentage. His formula is correct if the growth rate is given as a rate constant, but that is not the case in this problem. The parameter "k" in his expression is related to the percentage growth rate by: k = ln(1 + r/100), where r is the percent growth rate. You can see that his expression cannot be correct by using it for the case in which the growth rate is 100% per hour. In this case, the sample would double in size each hour. Using his expression, however, one would say that the sample size would be:

x(t) = x(0) * exp(1*t)

x(t) = x(0) * e

After 1 hour the sample would be "e" (~= 2.718) times as large. This is not correct.

If we use the formula relating k and r I gave above, however, the correct value of k would be

k = ln(1 + 100/100) = ln(2)

Then

x(t) = x(0)*exp(ln(2)*t)

x(t) = x(0)*2^t

after 1 hour,

x(1) = 2*x(0)

which is the correct result.

Answer 2

Hope this helps:) medal?!