Question

A cowboy at a dude ranch fills a horse trough that is 1.5 m long, 70 cm wide, and 35 cm deep. He uses a 2.1 cm diameter hose from which water emerges at 1.6 m/s. How long does it take him to fill the trough?

Answer 1

Length of the horse through L = 1.5 m

Width of the horse through W = 70 cm = 0.7 m

Depth of the horse through H = 35 cm = 0.35 m

Volume of the horse through V = L * W * H

= 1.5 × 0.7 × 0.35

= 0.3675 m³

Diameter of the hose is D = 2.1 cm = 0.021 m

Radius of the hose is R = D / 2 = 0.021 / 2

= 0.0105 m

= 1.05 × 10^-2 m

Hose is circular. So it's area A = R²

= 3.14 × ( 1.05 × 10^-2 )²

= 3.461 × 10^-4 m²

Speed of the water coming out of the hose is v = 1.6 m/s

Amount of the water flowing through the hose given by continuity equation.

So amount of water flowing through the hose is Area × speed = A × v

When the water falls into the horse through , it's volume is filled by water. Amount of water flowing into it will be equal to rate of change in its volume.

So we can write ,

Area × speed = volume / time

A × v = V / t

Time t = V / ( A × v )

= 0.3675 /( 3.461 × 10^-4 × 1.6 )

= ( 0.3675 × 10⁴ ) / ( 5.537 )

= 664.5569 seconds

Here we have used S I units , so we got answer in seconds.

If we want an answer in minutes , we have to divide it by 60 as 1 minute is equal to 60 seconds

Time t = 664.5569 / 60

= 11.0759 min

So it takes about 11 minutes to fill the horse through with the water