Final answer:
To find when the bacterial population exceeds 630 using the given equation, we need to solve for t in P(t)=350e^0.21t. The solution shows that the population will exceed 630 at approximately t=3.5 hours.
Step-by-step explanation:
The student asks to estimate when a population of bacteria that grows according to the equation P(t) = 350e0.21t will exceed 630. To solve this, we set up the equation 630 = 350e0.21t and solve for t.
We start by dividing both sides of the equation by 350, obtaining e0.21t = 630/350, which simplifies to e0.21t = 1.8.
To solve for t, we take the natural logarithm of both sides to get ln(e0.21t) = ln(1.8). This gives us 0.21t = ln(1.8) since the natural logarithm and the exponential function are inverses of each other.
Dividing by 0.21 on both sides gives us t = ln(1.8)/0.21. When you calculate that with a calculator, you get t ≈ 3.5.
Therefore, the bacterial population will exceed 630 at approximately 3.5 time units, where t is most likely measured in hours in this biological context.