A cylinder of radius 15 cm and height 24 cm is filled with water. A solid hemisphere of radius
7cm is submerged into the cylinder and removed. Find the change in height of water level in
the cylinder. (4 marks)
To find the change in height of the water level in the cylinder, we need to first calculate the volume of the solid hemisphere.
The volume of a solid hemisphere can be calculated using the formula:
V = (2/3)πr^3
Given that the radius of the solid hemisphere is 7 cm, we can substitute this value into the formula:
V = (2/3)π(7^3)
V = (2/3)π(343)
V = (2/3)(1078.47)
V ≈ 719.10 cm³
Since the solid hemisphere is submerged in the cylinder, the change in water level will be equal to the volume of the hemisphere.
Now, the volume of the cylinder can be calculated using the formula:
V = πr^2h
Given that the radius of the cylinder is 15 cm and the height is 24 cm, we can substitute these values into the formula:
V = π(15^2)(24)
V = π(225)(24)
V = π(5400)
V ≈ 16964.16 cm³
Therefore, the change in water level in the cylinder is approximately 719.10 cm³.
To find the change in height of the water level in the cylinder, we need to determine the volume of the hemisphere and the volume of the water displaced by the hemisphere.
1. Volume of the hemisphere:
The formula for the volume of a hemisphere is given by V = (2/3) * π * r^3, where r is the radius of the hemisphere. Substituting the given value, the volume of the hemisphere is:
V_hemisphere = (2/3) * π * (7^3)
V_hemisphere = (2/3) * π * 343
V_hemisphere = (686/3) * π cm^3
2. Volume of water displaced by the hemisphere:
The volume of water displaced by the hemisphere is equal to the volume of the hemisphere itself. Therefore, the volume of water displaced is:
V_displaced = V_hemisphere = (686/3) * π cm^3
3. Change in the height of the water level:
To find the change in height of the water level, we need to find the height of the water column that has the same volume as the displaced water. The formula to find the height of the water column is:
h = V_displaced / (π * r^2), where r is the radius of the cylinder.
Substituting the given values, the change in height of the water level is:
h = (686/3) * π / (π * 15^2)
h = 686 / (3 * 15^2)
h = 686 / (3 * 225)
h = 686 / 675
h ≈ 1.015 cm
Therefore, the change in height of the water level in the cylinder is approximately 1.015 cm.