A cowboy at a ranch fills a water trough that is 1.1 m long, 62 cm wide, and 48 cm deep. He uses a hose having a diameter of 2.4 cm, and the water emerges from the hose at 2.4 m/s.

How long does it take the cowboy to fill the trough?
Answer in units of s.

Volume tank= 1.1*.62*height

DV/dt= 1.1*.62 dh/dt

Hose: dv/dt= area*velocity

areahose*velocity= 1.1*.62*dh/dt

itegrate over time
areahose*velocity*time= 1.1*.62*h
solve for time.

Volume= l * W *h in meters

Velocity= 2.4 m/s
Area= (pi)(r^2)
Flow rate= Velocity times area
Time= Volume/Flow Rate

To find out how long it takes the cowboy to fill the trough, we need to calculate the volume of the trough and then divide it by the flow rate of the water.

Step 1: Calculate the volume of the trough
The volume of a rectangular prism (trough) is given by the formula V = length × width × height.

Given:
Length (l) = 1.1 m = 110 cm (since 1 m = 100 cm)
Width (w) = 62 cm
Depth (d) = 48 cm

V = 110 cm × 62 cm × 48 cm
V = 408,960 cm³

Step 2: Convert the volume to liters
Since the volume is in cubic centimeters (cm³), let's convert it to liters. 1 liter is equal to 1000 cm³.

Volume (V) in liters = 408,960 cm³ ÷ 1000
V = 408.96 liters

Step 3: Calculate the fill time
To calculate the fill time, we need to divide the volume by the flow rate.

Given:
Hose diameter = 2.4 cm
Water emerges from the hose at a speed of 2.4 m/s.

To find the flow rate, we can use the formula:
Flow rate (Q) = π × (diameter/2)² × velocity

Q = π × (2.4 cm/2)² × 2.4 m/s
Q ≈ 9.077 cm² × m/s

Since 1 liter is equal to 1000 cm³, we need to convert the flow rate from cm² × m/s to liters/s by dividing it by 1000.

Flow rate (Q) in liters/s = 9.077 cm² × m/s ÷ 1000
Q ≈ 0.009077 liters/s

Finally, we can calculate the fill time by dividing the volume by the flow rate:

Fill time = Volume / Flow rate
Fill time = 408.96 liters / 0.009077 liters/s

The units of liters cancel out, leaving us with seconds (s) as the unit:

Fill time ≈ 45087.69 s

Therefore, it takes the cowboy approximately 45087.69 seconds (or about 12.52 hours) to fill the trough.