Answer:
(a) P(t) = 200e^(ln(4)t/4)
(b) 4525.4834
(c) 5.42
Explanation:
(a) To express the population after t hours as a function of t, we need to find the constant of proportionality k in the differential equation dP/dt = kP, where P is the population and t is the time. We can use the given information that P(0) = 200 and P(4) = 800 to solve for k.
We have P(t) = Ce^(kt), where C is an arbitrary constant. Plugging in P(0) = 200, we get C = 200. Then, plugging in P(4) = 800, we get 800 = 200e^(4k), which simplifies to e^(4k) = 4. Taking the natural logarithm of both sides, we get 4k = ln(4), or k = ln(4)/4.
Therefore, the population after t hours as a function of t is P(t) = 200e^(ln(4)t/4).
(b) To find the population after 9 hours, we simply plug in t = 9 into the function P(t). We get P(9) = 200e^(ln(4)*9/4), which is approximately 4525.4834.
(c) To find how long it will take for the population to reach 1310, we need to solve for t when P(t) = 1310. We have 1310 = 200e^(ln(4)t/4), which simplifies to e^(ln(4)t/4) = 6.55. Taking the natural logarithm of both sides, we get ln(4)t/4 = ln(6.55), or t = 4ln(6.55)/ln(4). This is approximately 5.42 hours.