A hemispherical bowl with a 30-centimeter radius contains some water, which is 12 centimeters deep. Find the volume of the water.

To find the volume of the water in the hemispherical bowl, we need to find the volume of the hemisphere and then subtract the volume of the remaining hemisphere that is above the water level.

Step 1: Calculate the volume of the entire hemisphere.
The formula for the volume of a hemisphere is given as:
V = (2/3) * π * r^3
where V is the volume and r is the radius.

Substituting the given radius into the formula:
V = (2/3) * 3.14 * (30 cm)^3
V = (2/3) * 3.14 * 27000 cm^3
V ≈ 56520 cm^3

Step 2: Calculate the volume of the remaining hemisphere above the water level.
The height of the water is 12 cm, which means the remaining hemisphere above the water level will have a radius of (30 cm - 12 cm = 18 cm) since only the lower part of the hemisphere is filled with water.

Using the same formula as in Step 1:
V_remaining = (2/3) * 3.14 * (18 cm)^3
V_remaining = (2/3) * 3.14 * 5832 cm^3
V_remaining ≈ 12245 cm^3

Step 3: Calculate the volume of the water by subtracting the volume of the remaining hemisphere from the volume of the entire hemisphere:
V_water = V - V_remaining
V_water = 56520 cm^3 - 12245 cm^3
V_water ≈ 44275 cm^3

Therefore, the volume of the water in the hemispherical bowl is approximately 44275 cm^3.

To find the volume of the water in the hemispherical bowl, we need to calculate the volume of the hemisphere and subtract the volume of the empty space (the hemisphere minus the water).

Step 1: Calculate the volume of the hemisphere:
The formula to calculate the volume of a hemisphere is V = (2/3)πr^3, where V is the volume and r is the radius.
Substituting r = 30 centimeters into the formula:
V_hemisphere = (2/3)π(30)^3 = (2/3)π(27000) = 18000π cubic centimeters.

Step 2: Calculate the volume of the empty space in the hemisphere (i.e., the hemisphere minus the water):
The empty space is a smaller hemisphere with a radius of 30 centimeters minus a smaller hemisphere with a radius of 18 centimeters (30 cm - 12 cm). Let's call this volume V_empty.
Using the same formula as before, but with r = 18 centimeters:
V_empty = (2/3)π(18)^3 = (2/3)π(5832) = 3888π cubic centimeters.

Step 3: Calculate the volume of the water:
To find the volume of the water, we subtract the volume of the empty space from the volume of the hemisphere:
V_water = V_hemisphere - V_empty = 18000π - 3888π = 14112π cubic centimeters.

Therefore, the volume of the water in the hemispherical bowl is 14112π cubic centimeters, or approximately 44384.8 cubic centimeters.

check out the volume of a spherical cap:

https://en.wikipedia.org/wiki/Spherical_cap