# Math Help - Rate of change question

1. ## Rate of change question

An insect colony has an initial populatio0n of 1200 insects. After 20 hours, the population has increased to 5000, if it's growing exponentially, Find.
a) the average rate of change for the first 6 months
b) the rate of growth after 6 hours

Note: Use iterations for b

The only thing i don't know is how to come up with an equation from the word problem (i can figure out the rest, if i have the equation)

My teachers said you have to use logs to come up with the equation

2. Originally Posted by supersaiyan
An insect colony has an initial populatio0n of 1200 insects. After 20 hours, the population has increased to 5000, if it's growing exponentially, Find.
a) the average rate of change for the first 6 months
b) the rate of growth after 6 hours

Note: Use iterations for b

The only thing i don't know is how to come up with an equation from the word problem (i can figure out the rest, if i have the equation)

My teachers said you have to use logs to come up with the equation
well, questions like these are nice. there are already general equations that we know we will get when we derive them here. the equations you need to know are:

$P' = rP$

where $P'$ is the rate of change of the population, $r$ is the rate of growth, and $P$ is the current population.

from this equation, we use calculus (logs involved) to derive the following standard equation for exponential growth

$P(t) = P_0e^{rt}$

where $P(t)$ is the population at time $t$, $P_0$ is the initial population and $r$ is the rate of growth

the formula has the property that if $t_d$ is the time for the population to double, we have $rt_d = \ln 2$

(there are analogous formulas and properties for exponential decay)