At which rate the water is draining from the tank after 5 min.

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 3.8, Problem 18E

(a)

To determine

To find: At which rate the water is draining from the tank after 5 min.

Expert Solution

The rate at which the water is draining from the tank after 5 min is, V'(5)=218.75kg/m.

Explanation of Solution

The volume of the remaining water after t minutes is, V(t)=5000(1t40)2,0t40.

The rate of change of the volume with respect to time is,

V(t)=5000(1t40)2V'(t)=25000(1t40)(140)=250(40t40)

Substitute t = 5 in V'(t),

V'(5)=250(40540)=250(3540)=250(78)=218.75

Thus, the rate at which the water is draining from the tank after 5 min is, V'(5)=218.75kg/m.

(b)

To determine

To find: The rate at which the water is draining from the tank after 10 min.

Expert Solution

The rate at which the water is draining from the tank after 10 min is, V'(10)=187.5kg/m.

Explanation of Solution

From part (a), the rate of change of the volume with respect to time is, V'(t)=250(40t40).

Substitute t = 10 in V'(t),

V'(10)=250(401040)=250(3040)=250(34)=187.5

Thus, the rate at which the water is draining from the tank after 10 min is, V'(10)=187.5kg/m.

(c)

To determine

To find: The rate at which the water is draining from the tank after 20 min.

Expert Solution

The rate at which the water is draining from the tank after 20 minutes is, V'(20)=125kg/m.

Explanation of Solution

From part (a), the rate of change of the volume with respect to time is, V'(t)=250(40t40).

Substitute t = 20 in V'(t),

V'(20)=250(402040)=250(2040)=250(12)=125

Thus, the rate at which the water is draining from the tank after 20 min is, V'(20)=125kg/m.

(d)

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).

Expert Solution

The rate at which the water is draining from the tank after 40 min is, V'(40)=0kg/m.

The water is flowing out the fastest at t = 5 min. and the slowest at t = 40 min.

Explanation of Solution

From part (a), the rate of change of the volume with respect to time is, V'(t)=250(40t40).

Substitute t = 40 in V'(t),

V'(40)=250(404040)=250(040)=0

Thus, the rate at which the water is draining from the tank after 40 min is, V'(40)=0kg/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.

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