A spherical tank in a petro-chemical plant has a radius of 5m.
Give an expression which, when integrated from y=-r to
y=d-r ,will give the volume of liquid in the tank where r is
the radius and d is the depth of liquid.
If liquid is entering the tank at a rate of 1500 litres per second and
the depth is 3m how long will it take to fill the tank? Give your
answer in seconds.
The volume of a thin slice of water is the area of the slice times its thickness, dy
At a depth of d, with d<r, we have
the radius of the surface of the water is x = √(r^2-y^2)
v = ∫[-r,d-r] π(r^2-y^2) dy
Now you can answer the last part by subtracting the volume from the whole sphere's volume of 4/3 pi * 5^3 and dividing by 1500L/s = 1.5 m^3/s
To find the expression for the volume of liquid in the tank, we need to consider the equation of a sphere and the given values for radius (r) and depth (d).
The equation of a sphere is given by V = (4/3)πr³, where V represents the volume, and r represents the radius.
In this case, since the liquid is filled up to a certain depth (d), we need to find the expression for the volume of the portion of the sphere below that depth.
To do this, we will subtract the volume of the spherical cap (the portion above the liquid level) from the volume of the full sphere.
The volume of the spherical cap can be calculated using the formula V_cap = (1/3)πh²(3r - h), where h represents the height or depth of the liquid.
Therefore, the expression for the volume of liquid in the tank, when integrated from y = -r to y = d-r, will be:
Volume = (4/3)πr³ - (1/3)πh²(3r - h)
Substituting the given values r = 5m and d = 3m, the expression becomes:
Volume = (4/3)π(5)³ - (1/3)π(3)²(3(5) - 3)
Simplifying this expression, we get:
Volume = (4/3)π(125) - (1/3)π(9)(12)
Next, to calculate the time it takes to fill the tank, we need to consider the rate at which liquid is entering the tank, given as 1500 liters per second.
Since the volume is in cubic meters, we need to convert liters to cubic meters. There are 1000 liters in a cubic meter, so 1500 liters per second is equivalent to 1.5 cubic meters per second.
To find the time it takes to fill the tank, we can divide the volume of the tank by the rate at which liquid is entering:
Time = Volume / Rate
= (Volume in cubic meters) / (1.5 cubic meters per second)
Therefore, the final step is to calculate the time it takes to fill the tank:
Time = (Volume / 1.5) seconds
Substituting the volume expression we derived earlier, the final expression to calculate the time is:
Time = [(4/3)π(125) - (1/3)π(9)(12)] / 1.5 seconds
Simplifying this expression will give the time in seconds.