A population of bacteria has exponential growth and will double after 10 hours. Find the growth rate to the nearest hundredth.
p = Pi e^kt
2 = e^10 k
ln 2 = 10 k
.693 = 10 k
k = .0693
To find the growth rate of the bacteria population, we can use the formula for exponential growth:
P(t) = P(0) * e^(rt)
Where:
P(t) is the population size at time t
P(0) is the initial population size
r is the growth rate
t is the time period
In this case, we know that the population will double after 10 hours. This means that P(t) = 2 * P(0) and t = 10. Plugging these values into the equation, we get:
2 * P(0) = P(0) * e^(r * 10)
Simplifying the equation, we can divide both sides by P(0):
2 = e^(10r)
To solve for r, we can take the natural logarithm (ln) of both sides:
ln(2) = ln(e^(10r))
Using the property of logarithms, we can bring down the exponent:
ln(2) = 10r * ln(e)
Since the natural logarithm of e is equal to 1, the equation simplifies to:
ln(2) = 10r
Now, we can solve this equation for r by dividing both sides by 10:
r = ln(2) / 10
Calculating this value, we find:
r ≈ 0.0693
Therefore, the growth rate of the bacteria population is approximately 0.0693, rounded to the nearest hundredth.