Answer :
Final answer:
The number of bacteria at time 0 is 500. It takes approximately 1.44 days for the number of bacteria to reach 2000.
Explanation:
The given exponential function for the number of bacteria at time t is: N(t) = 500e^(0.04t).
To find the number of bacteria at time 0, substitute t = 0 into the function:
N(0) = 500e^(0.04(0)) = 500e^0 = 500(1) = 500 bacteria.
To find how many days it takes for the number of bacteria to reach 2000, set N(t) = 2000 and solve for t:
2000 = 500e^(0.04t)
e^(0.04t) = 2000/500 = 4
Take the natural logarithm of both sides to isolate the exponent:
ln(e^(0.04t)) = ln(4)
0.04t = ln(4)
t = ln(4)/0.04
Using a calculator, t ≈ 34.66 hours or approximately 1.44 days.
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