Water is draining from a swimming pool in such a way that the remaining volume of water after t minutes is V = 200(50 - t)^2 cubic meters. Find :

(a) the average rate at which the water leaves the pool in the first 5 minutes.

(b) Find the instantaneous rate at which the water is leaving at t = 5 minutes.

a)
V(0) = 500,000
V(5) = 405,000
(V(5)-V(0))/(5-0) = -19,000 ft^3/min

Can someone help me on question b and check if question a is correct

a) is correct

b)
dV/dt = 400(50-t)^1 (-1)
at t = 5
dV/dt = -400(45) = -18,000 ft^3/min

how did you get 400?

400 = 200*2

Sure! Let's first verify the answer to question (a). To find the average rate at which the water leaves the pool in the first 5 minutes, we need to calculate the change in volume over 5 minutes and divide it by the time.

Given that V = 200(50 - t)^2, we can substitute t = 0 into the equation to find V(0):
V(0) = 200(50 - 0)^2
V(0) = 200(50^2)
V(0) = 500,000 cubic meters

Next, substitute t = 5 into the equation to find V(5):
V(5) = 200(50 - 5)^2
V(5) = 200(45^2)
V(5) = 405,000 cubic meters

Now, we can calculate the average rate of change using the formula:
Average rate = (V(5) - V(0)) / (5 - 0)
= (405,000 - 500,000) / 5
= -95,000 / 5
= -19,000 cubic meters per minute

So, your answer for question (a) is correct.

Now, let's move on to question (b) and find the instantaneous rate at which the water is leaving at t = 5 minutes. The instantaneous rate can be determined by finding the derivative of the volume function with respect to time (t).

Differentiating the volume function V(t) = 200(50 - t)^2, we get:
dV/dt = 2 * 200 * (50 - t) * (-1)
= -400(50 - t)

Now, to find the instantaneous rate at t = 5 minutes, substitute t = 5 into the derivative expression:
dV/dt |(t=5) = -400(50 - 5)
= -400(45)
= -18,000 cubic meters per minute

Therefore, the instantaneous rate at which water is leaving the pool at t = 5 minutes is -18,000 cubic meters per minute.

If you have any further questions, feel free to ask!

Do you not know how to find the derivative using the chain rule?

If not, this is not a forum to teach it to you from scratch.