The growth of a colony of bacteria is given by the equation:
Q = Q₀ * e^(0.195t)
where:
Q₀ = initial number of bacteria
t = time in hours
Q = number of bacteria at time t
Let's calculate the number of bacteria after half a day, which is 12 hours:
Q = 500 * e^(0.195 * 12)
Using a calculator, we can evaluate this expression:
Q ≈ 500 * e^(2.34)
Q ≈ 500 * 10.397
Q ≈ 5198.5
So, after half a day (12 hours), there are approximately 5198.5 bacteria in the colony.
Next, let's determine how long it will take to reach a bacteria population of 10,000 in the colony:
Q = 10000
500 * e^(0.195t) = 10000
Dividing both sides by 500:
e^(0.195t) = 10000 / 500
e^(0.195t) = 20
Taking the natural logarithm (ln) of both sides:
0.195t = ln(20)
Now, we solve for t:
t = ln(20) / 0.195
Using a calculator:
t ≈ 6.207
So, it will take approximately 6.207 hours to reach a bacteria population of 10,000 in the colony.