Explanation:
The rate of change of the population is proportional to the population, so we can use the exponential growth formula:
P(t) = P0 * e^(kt)
Where P0 is the initial population (300), k is the constant of proportionality, and t is the time in hours. We can use the data from t=24 hours to solve for k:
1000 = 300 * e^(24k)
Dividing both sides by 300 and taking the natural logarithm of both sides:
ln(1000/300) = 24k
ln(3.33) = 24k
k = ln(3.33) / 24
Now that we have k, we can plug it into the formula and solve for t when P(t) = 500:
P(t) = 300 * e^(kt)
500 = 300 * e^(kt)
e^(kt) = 500/300
e^(kt) = 5/3
kt = ln(5/3)
t = ln(5/3) / k
Substituting k = ln(3.33) / 24:
t = ln(5/3) / (ln(3.33) / 24)
t ≈ 11.7 hours.