Pre Calculus Word Problem Help?

Question

If a colony of bacteria has a population of 3000 at 12 noon and a population of 5000 at 3 pm calculate the growth constant r and find the time it will take to have the colony grow to 11,000

Answer 1

This is exponential growth: p(t) = Ae^kt
(colton's answer is wrong, it's for linear growth, i.e. new bacteria aren't allowed to reproduce)

where we choose t0 as 12 noon, and measure time in hours.

Then p(0) = 3000 => A = 3000
p(3) = 5000 => Ae^3k = 5000
=> 3k = ln(5/3)
=> k = 1/3 * ln(5/3)

then p(T) = 3000e^(t*1/3 * ln(5/3))

Solve p(T) = 11,000

p(t) = 3000e^(T*1/3 * ln(5/3))
= 11,000

3000e^(T*1/3 * ln(5/3)) = 11,000
T*1/3 * ln(5/3) = ln(11/3)
T = 3ln(11/3)/ln(5/3)
T = 7.63 hrs = 7hr 38min

Answer 2

5000 = 3000r^3
5/3 = r^3
r = cube root of 5/3 = 1.1856

but you can get the time to grow to 11,000 without ever knowing r:

5/3 = r^3
log (5/3) = 3 log r
11/3 = r^t
log (11/3) = t log r
divide 2nd and 4th equations:

log (5/3) ...... 3
------------ = --------
log (11/3) ..... t

the log r on the right divides out. so

t = [3 log (11/3)] / log (5/3) = 7.63 hrs after noon.