for 0<=t<=21 the rate of change of the number of blakc 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?
R = 3√t cos(t/3)
R' = 3/2√t cos(t/3) - √t sin(t/3)
R' = 0 when
3/2√t cos(t/3) = √t sin(t/3)
3cos(t/3) = 2t sin(t/3)
3/2 cot(t/3) = t
t = 1.96, 9.88, 19.08
not sure how there are 500 flies at t=0. That doesn't fit R(t). Anyway, if that can be fixed, just plug in those values for t to get what you need.
wait..dont u hve to integrate it?
oops. yes. I misread the problem. Didn't see the "rate of change" phrase.
wait..so how would you integrate that equation..im hving trouble with tht..
Beats me. It doesn't use standard elementary functions. Are you studying numerical methods?
To find the maximum number of flies on the island, we need to find the maximum value of the rate of change function R(t) for the given time interval 0 ≤ t ≤ 21.
First, let's find the derivative of R(t) with respect to t. The derivative of R(t) will give us the rate at which the rate of change function is increasing or decreasing.
R'(t) = dR(t)/dt
Using the chain rule, we can find the derivative of R(t):
R'(t) = (d/dt) [3√(t)cos(t/3)]
To simplify this expression, we can use the product rule and the chain rule:
R'(t) = 3/2 * t^(-1/2) * cos(t/3) - √(t) * sin(t/3) * (1/3)
Next, we need to find the critical points of R'(t) within the given interval 0 ≤ t ≤ 21. Critical points occur where the rate of change is either zero or undefined.
Setting R'(t) = 0, we get:
3/2 * t^(-1/2) * cos(t/3) - √(t) * sin(t/3) * (1/3) = 0
Multiplying through by 6t^(3/2), we get:
9 * cos(t/3) - 2√(t) * sin(t/3) = 0
Now, to find the maximum point, we need to check both the endpoints of the interval (t=0 and t=21) and the critical points where R'(t) = 0.
At t=0, the rate of change is R'(0) = 3/2 * 0^(-1/2) * cos(0/3) - √(0) * sin(0/3) * (1/3) = undefined.
So, we only need to consider the critical points.
Unfortunately, finding the exact solutions to the equation 9 * cos(t/3) - 2√(t) * sin(t/3) = 0 is not straightforward. We can approach this by using numerical methods like graphing, a graphing calculator, or computer software.
Using a graphing calculator or software, we can plot the graph of R'(t) and find the maximum value within the interval 0 ≤ t ≤ 21.
Once we have the maximum value of R'(t), say R'(t_max), we need to find the corresponding value of t, which gives us the maximum number of flies.
To find the maximum number of flies, we need to add the initial number of flies (500) to the maximum value of the rate of change:
Maximum number of flies = 500 + R'(t_max)
To find the nearest whole number, you can round the result to the nearest whole number.
Note: Since finding the exact critical points is not feasible without numerical methods, I cannot provide you with an exact answer. However, by following the steps provided, you should be able to find an approximate value for the maximum number of flies on the island.