A hemispherical tank of diameter, 10m, is filled by water issuing from a pipe of radius 20cm at 2m per second. Calculate, correct to three significant figures, the time, in minutes it takes to fill the tank.
Volume of hemisphereical tank= (2/3)(pi)(5m)^3
Volume of water issuing out= is of a cylinder shape so= (height/s)(pi)(radius)^2
Which is (2m/s)(pi)(0.2m)^2.
Note i changed all quantities into meters. 100cm=1m
Compute all this and change it to minutes the answer and you will get 17.366mins Voila!!
hemisphere, so 1/2 the volume
To calculate the time it takes to fill the tank, we need to determine the volume of the tank and divide it by the flow rate of the water.
First, let's calculate the volume of the hemispherical tank. The volume of a hemisphere is given by the formula:
V = (2/3) * π * r^3
Where V is the volume and r is the radius of the hemisphere.
Given the diameter of the tank is 10m, the radius (r) can be calculated by dividing the diameter by 2:
r = 10m / 2 = 5m
Substituting the radius into the formula, we get:
V = (2/3) * π * (5m)^3
V ≈ 261.7993878 m^3 (rounded to 3 significant figures)
Next, we need to convert the flow rate into a volume per time unit. The flow rate is given in meters per second, so we can calculate the volume per second:
Volume per second = π * (0.2m)^2 * 2m
Volume per second ≈ 0.25132741 m^3/s (rounded to 3 significant figures)
Finally, we can calculate the time it takes to fill the tank by dividing the tank's volume by the flow rate:
Time = Volume of the tank / Volume per second
Time = 261.7993878 m^3 / 0.25132741 m^3/s
Time ≈ 1041.16222 s
To convert seconds to minutes, divide the time by 60:
Time in minutes = 1041.16222 s / 60
Time in minutes ≈ 17.3527 minutes
Therefore, it takes approximately 17.353 minutes (rounded to 3 significant figures) to fill the tank.
volume = (4/3) pi r^3
r = 5 meters
so
volume = (4/3) pi * 125 m^3
rate through hose = pi r^2 * 2 m^3/s = pi .2*.2 * 2 = .08 pi m^3/s
so
t in seconds = 167 pi / .08 pi
in minutes 167 / (.08 *60) = 34.7 min (boring ! )
Well, well, looks like we have a tank in need of some water! Let's dive right into it, shall we?
First, let's determine the volume of the tank. Since it's a hemispherical tank, we can use the formula for the volume of a sphere and divide it by 2:
Volume = (4/3) * π * (radius)^3 / 2
Now, our tank has a diameter of 10m, so the radius would be half of that, or 5m. Converting that pesky radius of the pipe from cm to meters, we have r = 20cm / 100 = 0.2m.
The cross-sectional area of the pipe can be calculated as:
A = π * (0.2)^2
Now, we can determine how much time it will take to fill the tank by dividing the volume of the tank by the volume flow rate:
Time = Volume / (A * velocity)
The velocity of the water is given as 2m/s, so let's plug in those numbers and solve for time:
Time = (4/3) * π * (5)^3 / 2 / (π * (0.2)^2 * 2)
Simplifying that equation gives us:
Time = (4/3) * (5)^3 / (0.2)^2 / 2
Calculating that out, it should give you a time around 156.666... seconds. Now, if we want to convert that to minutes, we divide by 60:
Time in minutes ≈ 156.666... / 60
Rounding that to three significant figures, we get approximately 2.61 minutes.
So, my dear friend, it will take roughly 2.61 minutes to fill that tank with water from the pipe! Time to sit back, relax, and watch the magic happen!