A water hose 1.42 cm in diameter is used to fill a 19.5 L bucket. If it takes 1.37 min to fill the bucket, what is
the speed at which the water leaves the hose? One liter is equal to 1000 cm3
.
To find the speed at which the water leaves the hose, we need to first calculate the volume of water that flows out of the hose in 1.37 minutes.
The volume of the bucket is given as 19.5 L. We know that 1 liter is equal to 1000 cm^3. So, the volume of the bucket in cm^3 is calculated as:
19.5 L * 1000 cm^3/L = 19500 cm^3
Next, we need to calculate the volume of water that flows out of the hose in 1.37 minutes. Since the diameter of the hose is given in cm, the radius is calculated as half the diameter:
Radius = 1.42 cm / 2 = 0.71 cm
The area of the hose's cross-section can be calculated using the formula for the area of a circle:
Area = π * radius^2
Area = 3.14 * (0.71 cm)^2 = 1.59 cm^2
To find the volume of water flowing from the hose per minute, we need to multiply the cross-sectional area by the speed at which the water flows out of the hose. Let's denote this speed as v.
Volume flow rate = Area * v
The volume flow rate is equal to the volume of water that flows out of the hose in 1.37 minutes, which is:
Volume flow rate = 19500 cm^3 / 1.37 min = 14233.58 cm^3/min
Substituting in the values, we can solve for v:
14233.58 cm^3/min = 1.59 cm^2 * v
Dividing both sides by 1.59 cm^2:
v = 14233.58 cm^3/min / 1.59 cm^2 ≈ 8950 cm/min
Therefore, the speed at which the water leaves the hose is approximately 8950 cm/min.
To find the speed at which the water leaves the hose, we need to determine the volume of water leaving the hose per unit of time.
First, we need to calculate the cross-sectional area of the hose.
Given:
- Diameter of the hose = 1.42 cm
We can calculate the radius (r) of the hose using the formula:
r = diameter / 2
= 1.42 cm / 2
= 0.71 cm
Next, we can calculate the cross-sectional area (A) of the hose using the formula for the area of a circle:
A = π * r^2
= π * (0.71 cm)^2
≈ 1.586 cm^2
Since the area is in square centimeters (cm^2), we need to convert it to square meters (m^2) to match the units of the volume. There are 10,000 cm^2 in 1 m^2, so:
A = 1.586 cm^2 * (1 m^2 / 10,000 cm^2)
≈ 0.0001586 m^2
Next, we need to determine the volume of water leaving the hose per unit of time. Given that it takes 1.37 minutes to fill a 19.5 L (19,500 cm^3) bucket, we can calculate the volume flow rate (Q) using the formula:
Q = volume / time
= (19,500 cm^3) / (1.37 min)
= 14,233.58 cm^3/min
Since we want the speed at which the water leaves the hose, we need to convert the volume flow rate from cm^3/min to m^3/s. There are 1,000,000 cm^3 in 1 m^3 and 60 seconds in 1 minute, so:
Q = 14,233.58 cm^3/min * (1 m^3 / 1,000,000 cm^3) * (1 min / 60 s)
= 0.00023722635 m^3/s
Finally, to calculate the speed at which the water leaves the hose, we divide the volume flow rate by the cross-sectional area:
Speed = Q / A
= 0.00023722635 m^3/s / 0.0001586 m^2
≈ 1.496 m/s
Therefore, the speed at which the water leaves the hose is approximately 1.496 m/s.