Question 1122273
you first need to find the exponential growth rate.


the formula for that is f = p * e^(rt)


f is equal to 4.5 * 10^6
p is equal to 1.5 * 10^6
t is equal to 3


you are solving for r.


formula becomes 4.5 * 10^6 = 1.5 * 10^6 * e^(3r)


divide both sides of this formula by 1.5 * 10^6 to get:


4.5 * 10^6 / (1.5 * 10^6) = e^(3r)


take the natural log of both sides of this formula and simplify to get:


ln(3) = ln(e^3r)


since ln(e^3r) is equal to 3r * ln(e) and ln(e) is equal to 1, this equation becomes:


ln(3) = 3r


solve for r to get:


r = ln(3) / 3


this results in r = .3662040962


that's your hourly exponential growth rate.


to see if this is good, take 1.5 * 10^6 and multiply it by e^(.3662040962 * 3).


you will get 4.5 * 10^6 which is exactly what you want, assuming the rate is calculated correctly, as it is.


to find out when the population will reach 8.0 * 10^6, use the same formula of f = p * e^(rt).


if you are starting from 1.5 * 10^6, the formula becomes:


8.0 * 10^6 = 1.5 * 10^6 * e^(.3662040962 * t)


divide both sides of the equation by 1.5 * 10^6 to get:


8.0 * 10^6 / (1.5 * 10^6) = e^(.3662040962 * t)


take the natural log of both sides and simplify to get:


ln(5 + 1/3) = .3662040962 * t


solve for t to get:


t = ln(5 + 1/3) / .3662040962 = 4.571157043


to confirm, take 1.5 * 10^6 and multiply it by e^(.3662040962 * 4.571157043).


you will get 8.0 * 10^6, as you should.


if you had started from 4.5 * 10^6, the formula would have become:


8.0 * 10^6 = 4.5 * 10^6 * e^(.3662040962 * t)


in that case, you would have solved for t to get:


t = ln(8.0 * 10^6 / (4.5 * 10^6) / .3662040962, resulting in:


t = 1.571157043


add that to the 3 hours to get from 1.5 * 10^6 to 4.5 * 10^6 and the total hours is 4.571157043.


the total hours to get from 1.5 to 4.5 million = 3.


the total hours to get from 4.5 to 8 million = 1.571157043


the total hours to get from 1.5 to 8 million = 4.571157043


the formula for this problem can be graphed as shown below:


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