To determine
To find: At which rate the water is draining from the tank after 5 min.
Answer
The rate at which the water is draining from the tank after 5 min is, V'(5)=−218.75 kg/m.
Explanation
The volume of the remaining water after t minutes is, V(t)=5000(1−t40)2,0≤t≤40.
The rate of change of the volume with respect to time is,
V(t)=5000(1−t40)2V'(t)=2⋅5000(1−t40)(−140)=−250(40−t40)
Substitute t = 5 in V'(t),
V'(5)=−250(40−540)=−250(3540)=−250(78)=−218.75
Thus, the rate at which the water is draining from the tank after 5 min is, V'(5)=−218.75 kg/m.
To determine
To find: The rate at which the water is draining from the tank after 10 min.
Answer
The rate at which the water is draining from the tank after 10 min is, V'(10)=−187.5 kg/m.
Explanation
From part (a), the rate of change of the volume with respect to time is, V'(t)=−250(40−t40).
Substitute t = 10 in V'(t),
V'(10)=−250(40−1040)=−250(3040)=−250(34)=−187.5
Thus, the rate at which the water is draining from the tank after 10 min is, V'(10)=−187.5 kg/m.
To determine
To find: The rate at which the water is draining from the tank after 20 min.
Answer
The rate at which the water is draining from the tank after 20 minutes is, V'(20)=−125 kg/m.
Explanation
From part (a), the rate of change of the volume with respect to time is, V'(t)=−250(40−t40).
Substitute t = 20 in V'(t),
V'(20)=−250(40−2040)=−250(2040)=−250(12)=−125
Thus, the rate at which the water is draining from the tank after 20 min is, V'(20)=−125 kg/m.
To determine
To find: The rate at which the water is draining from the tank after 40 min; at what time is the water is draining out fastest and the slowest; summarize the results obtained from the parts (a), (b), (c) and (d).
Answer
The rate at which the water is draining from the tank after 40 min is, V'(40)=0 kg/m.
The water is flowing out the fastest at t = 5 min. and the slowest at t = 40 min.
Explanation
From part (a), the rate of change of the volume with respect to time is, V'(t)=−250(40−t40).
Substitute t = 40 in V'(t),
V'(40)=−250(40−4040)=−250(040)=0
Thus, the rate at which the water is draining from the tank after 40 min is, V'(40)=0 kg/m.
From the above sub parts, it is identified that the water flow is very fast at t = 5 min. and is too slow at t = 40 min.
Also, notice that at t = 40 min. the velocity is 0 and hence the water flow is very slow.