Answer:
approximately 10.3 days to reach 500 bacteria
Explanation:
Let's use the formula for exponential growth to write an equation that represents the situation: N = N0 * e^(rt)
where N is the number of bacteria, N0 is the initial number of bacteria, e is Euler's number (approximately 2.718), r is the growth rate, and t is the time in days.
We know that after 4 days, the number of bacteria is 71, so we can plug these values into the equation to solve for the growth rate: 71 = N0 * e^(4r)
Similarly, after 7 days (4 + 3), the number of bacteria is 185: 185 = N0 * e^(7r)
Now we have two equations with two unknowns (N0 and r). We can divide the second equation by the first equation to eliminate N0: 185/71 = e^(3r)
Taking the natural logarithm of both sides, we get: ln(185/71) = 3r
Solving for r, we get: r = ln(185/71) / 3 ≈ 0.558
Now we can use the first equation and the growth rate we just found to solve for N0:
71 = N0 * e^(4 * 0.558)
N0 ≈ 11.7
So the initial number of bacteria was approximately 11.7.
To find out how long it will take to reach 500 bacteria, we can plug in the values we know into the equation and solve for t: 500 = 11.7 * e^(0.558t)
Dividing both sides by 11.7, we get: e^(0.558t) ≈ 42.74
Taking the natural logarithm of both sides, we get: 0.558t ≈ ln(42.74)
Solving for t, we get: t ≈ ln(42.74) / 0.558 ≈ 10.3 days
Therefore, it will take approximately 10.3 days for the sandwich to be fully covered with bacteria.