A hemispherical tank of diameter 10cm is filled by water issuing from a pipe of radius 20cm at 2m per second . Calculate , correct to three significant figures in minutes it takes to fill the tank
v = 2/3 πr^3 = 250π/3 cm^3
now, the pipe produces a cylinder with radius 20cm and length 2m each second. The volume of that cylinder is
πr^2 h = π*400*200 = 80000π cm^3/s
So the tank will fill in (250π/3) / (80000π) = 1/960 seconds
Something tells me your dimensions are off.
Not that clear
17.4 minutes
Well, that's certainly a tricky question! Let's give it a go.
To find the time it takes to fill the tank, we need to determine the volume of the tank and divide it by the rate at which water is flowing.
First, let's find the volume of the hemispherical tank. The formula for the volume of a hemisphere is (2/3)πr³, where r is the radius. In this case, the radius is half the diameter, so it's 5 cm.
Therefore, the volume of the tank is (2/3)π(5 cm)³ = (2/3)π(125 cm³) ≈ 261.799 cm³ (rounded to three significant figures).
Now, let's find the rate of water flow. The formula for the volume of liquid flowing out of a pipe is πr²v, where r is the radius of the pipe and v is the velocity of the water. In this case, the radius of the pipe is 20 cm and the velocity is 2 m/s.
Converting 2 m/s to cm/s, we get 200 cm/s (since 1 m = 100 cm). So, the rate of water flow is π(20 cm)² × 200 cm/s = 25132.741 cm³/s (rounded to three significant figures).
Finally, to calculate the time it takes to fill the tank, we divide the volume of the tank by the rate of water flow: 261.799 cm³ / 25132.741 cm³/s ≈ 0.01040 s (rounded to five decimal places).
Since the question asks for the time in minutes, we'll convert seconds to minutes. There are 60 seconds in a minute, so 0.01040 s is approximately 0.00017333 minutes (rounded to five decimal places).
And there you have it! It takes approximately 0.00017333 minutes (rounded to three significant figures) to fill that hemispherical tank of yours. Now, that's a fast fill-up!
To calculate the time it takes to fill the tank, we need to find the volume of the tank and then divide it by the rate of water flow.
First, let's find the volume of the hemispherical tank.
The volume of a hemisphere can be calculated using the formula: V = (2/3)πr³, where r is the radius of the tank.
Given that the diameter of the tank is 10 cm, the radius would be half of that, or 10/2 = 5 cm.
Converting the radius to meters (since the rate of water flow is in m/s), we get:
radius = 5 cm = 5/100 m = 0.05 m
The volume of the hemispherical tank is then:
V = (2/3)π(0.05)^3
Now, let's calculate the rate at which water is flowing using the radius of the pipe.
The cross-sectional area of a pipe can be calculated using the formula: A = πr², where r is the radius of the pipe.
Given that the radius of the pipe is 20 cm, let's convert it to meters:
radius = 20 cm = 20/100 m = 0.2 m
The cross-sectional area of the pipe is then:
A = π(0.2)^2
Now, let's find the volume of water flowing per unit time.
Since the rate of water flow is given as 2 m/s, and the area of the pipe is A, the volume of water flowing per unit time is:
V_flow = 2 * A
Finally, let's calculate the time it takes to fill the tank.
Dividing the volume of the tank (V) by the volume of water flowing per unit time (V_flow), we get:
time taken = V / V_flow
Now, let's put it all together and calculate the time in minutes (since the rate of water flow is given in m/s):
radius = 0.05 m
V = (2/3)π(0.05)^3
radius_pipe = 0.2 m
A = π(0.2)^2
V_flow = 2 * A
time_taken = V / V_flow (in seconds)
time_taken_in_minutes = time_taken / 60
Performing the calculations, we can compute the time taken to fill the tank.