A metal sphere whose diameter is 150cm is lowered into the vessel till it rests on the base. Calculate, in terms of pi, the volume of water which overflows in cubic metres. The vessel has height and diameter of 2m, is a cylinder without a lid and stands horizontally, containing water to a depth of 150cm
No water escapes.
Even if the sphere were a cylindrical plug with diameter of 2m and a height of 150cm, it would only raise the water level by 150 cm, leaving it well below the top of the cylinder, at 2m high.
Is there a typo? (I mean, aside from something which stands "horizontally".)
To find the volume of water that overflows, we first need to determine the volume of the metal sphere and the volume of the vessel.
1. Volume of the metal sphere:
The volume of a sphere can be calculated using the formula V = (4/3) * π * r³, where r is the radius of the sphere.
In this case, the diameter of the sphere is given as 150 cm, which means the radius is half the diameter, or 150 cm / 2 = 75 cm.
Converting the radius to meters, we have 75 cm / 100 = 0.75 m.
Now we can calculate the volume of the sphere:
V_sphere = (4/3) * π * (0.75)^3
2. Volume of the vessel:
The vessel is a cylinder without a lid, so its volume can be calculated using the formula V = π * r² * h, where r is the radius of the base and h is the height of the cylinder.
In this case, the diameter of the vessel is given as 2 m, which means the radius is half the diameter, or 2 m / 2 = 1 m.
The height of the vessel is given as 2 m.
Now we can calculate the volume of the vessel:
V_vessel = π * (1)^2 * 2
3. Calculate the volume of water overflow:
Since the metal sphere is lowered into the vessel until it rests on the base, the volume of water that will overflow is equal to the difference between the volume of the sphere and the volume of the vessel:
V_overflow = V_sphere - V_vessel
Now you can substitute the calculated values into the above equations to find the volume of water that overflows in terms of pi.