for 0<=t<=21 the rate of change of the number of black flies on a coastal island at time t days is modeled by R(t)=3sqrt(t)cos(t/3) flies per day. There are 500 flies on the island at the time t=0. To the nearest whole #, what is the max # of flies for 0<=t<=21? plz solve this completely b/c the choices r: 1)500 2)510 3)520 4)530 5)540
dN/dt = 3*t^1/2*cos(t/3)
Use separation of variables.
dN = 3*t^1/2*cos(t/3)*dt
Integrate both sides.
N - N(t=0) = Integral of
3*t^1/2*cos(t/3)*dt, from t=0 to t
I don't have time to finish this. I already explained in a previous answer how to get the time of maximum fly population.
is it from 0 to 21? or 0 to 4.7??
The maximum fly population is at t = 3 pi/2 = 4.712 days. Integrate R to that limit.
The R function varies as follows: 0 at t=0, 2.835 at t=1, 3.334 at t=2, 2.807 at t=3, 1.411 at t=4 and 0 at t=4.712 The integral of R from 0 to 4.7 will be about ten. That would make the answer 510. Not much of a change.
To find the maximum number of black flies on the island for the given time interval, we need to find the maximum value of the rate function R(t) and then add it to the initial number of flies. Here's how we can solve it step-by-step:
1. Calculate the rate of change of the number of black flies for each value of t within the given interval:
R(t) = 3√(t) * cos(t/3)
2. To find the maximum value of R(t), we need to find the critical points where the derivative equals zero or is undefined. Let's find the derivative of R(t):
R'(t) = (3/2)√(t) * cos(t/3) - (1/2)√(t) * sin(t/3)
3. Now, set the derivative equal to zero and solve for t:
(3/2)√(t) * cos(t/3) - (1/2)√(t) * sin(t/3) = 0
Simplifying the equation, we get:
3√(t) * cos(t/3) = √(t) * sin(t/3)
Dividing both sides by √(t), we have:
3cos(t/3) = sin(t/3)
4. From the given time interval (0 ≤ t ≤ 21), use numerical methods or a graphing calculator to find the values of t where the derivative is zero or undefined. These values will give us potential maxima and minima of R(t).
5. Evaluate R(t) at each critical point and the endpoints of the interval to find the maximum value:
- Substitute each critical point into R(t) and record the corresponding values.
- Evaluate R(0) and R(21) and record those values as well.
6. Compare all the values obtained in step 5 and identify the maximum number of flies.
Let's perform these steps to find the answer.