If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes.

If A Tank Holds 5000 Gallons

After time t, the following is the relation that represents the volume V of water that remains in the tank as per Torricelli’s Law.5000(1t40)2=V,  where 0t40

Volume

Volume

As the water is draining from the tank, calculate its rate after (a)5min and (b)10min.

Time

Time

Also, find the time at which the rate of water draining from the tank is fastest and slowest.

The aim of this article is to find the rate of water draining from the tank at a certain instance of time and find the time of fastest and slowest drain rate.

The basic concept behind this article is the use of Torricelli’s Equation to calculate the rate of flow.

The Rate of Flow of a given volume V is calculated by taking the first derivative of Torricelli’s Equation with respect to time t.

Rate of Flow=ddt(Torricellis Equation for Volume)=ddt(V)

Torricellis law

Torricelli’s Law.

Expert Answer

Given that:

Torricelli’s Equation for the Volume of Water remaining in the Tank is:

5000(1t40)2=V,  where 0t40

To calculate the rate at which water is draining at different instances of time t, we will be taking the first derivative of Torricelli’s Equation with respect to time t.

ddt(V)=ddtV(t)

ddtV(t)=ddt[5000(1t40)2]

V(t)=5000×2(1t40)×(140)

V(t)=250(1t40)

The negative sign indicates that the rate at which the water is draining is decreasing with time.

To calculate the rate at which water is draining from the tank after 5min, substitute t=5 in the above equation:

V(5)=250(1540)

V(5)=218.75GallonsMin

To calculate the rate at which water is draining from the tank after 10min, substitute t=10 in the above equation:

V(10)=250(11040)

V(10)=187.5GallonsMin

To calculate the time at which rate of water draining from the tank is fastest or slowest, take the following assumptions from the given minimum and maximum range of t

1st Assumption t=0 min

2nd Assumption t=40 min

For 1st assumption of t=0

V(0)=250(1040)

V(0)=250GallonsMin

For 2nd assumption of t=40

V(40)=250(14040)

V(40)=0GallonsMin

Hence, it proves that the rate at which the water is draining is fastest when V(t) is maximum and slowest when V(t) is minimum. Thus, the fastest rate at which the water is draining is at the start when t=0min and the slowest at the end of the drain when t=40min. As the time passes, the rate of drain becomes slower until it becomes 0 at t=40min

Numerical Result

The rate at which water is draining from the tank after 5min is:

V(5)=218.75GallonsMin

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

V(10)=187.5GallonsMin

The fastest rate of the drain is at the start when t=0min and the slowest at the end when t=40min.

Example

Water is draining from a tank containing 6000 gallons of water. After time t, the following is the relation that represents the volume V of water that remains in the tank as per Torricelli’s Law.

6000(1t50)2=V,  where 0t50

Calculate its rate of drain after 25min.

Solution

ddtV(t)=ddt [ 6000(1t50)2 ]

V(t)=240(1t50)

To calculate the rate at which water is draining from the tank after 25min, substitute t=5 in the above equation:

V(t)=240(12550)

V(t)=120GallonsMin

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