In an experiment, the population P of bacteria is given by P=1000e^kt,where k is constant.If the size of population bacteria is found to be 3000 after 10 hours, what is the size of the population after 5 hours?Show me the solution please.
3000 = 1000 e^10 k
3 = e^10k
ln 3 = 10 k
solve for k
n = 1000 e^5k with same k
do e^5k on your calculator
To find the size of the population after 5 hours using the given formula P=1000e^kt, we can use the information given about the population size after 10 hours and solve for the value of k.
We know that after 10 hours, the population size is 3000, so we can substitute the values into the formula:
P = 1000e^(10k) = 3000
Now we can solve for k:
e^(10k) = 3000/1000 = 3
Taking the natural logarithm ln on both sides:
ln(e^(10k)) = ln(3)
Simplifying, we get:
10k = ln(3)
Now, solve for k:
k = ln(3)/10
With the value of k, we can find the size of the population after 5 hours by substituting it into the formula:
P = 1000e^(5k)
P = 1000e^(5 * (ln(3)/10))
Calculating this expression will give us the size of the population after 5 hours.