A hemispherical bowl with a 30-cm radius has some water in it. The water is 15 cm deep.
Find the volume of water, to the nearest cubic cm.
What if the height of the water is only 15 cm, half the height of the hemisphere?
5625π cm³
Solution:
V = ⅓πd²(3r - d)
d is the depth and r is the radius
To find the volume of water in the hemispherical bowl, we need to calculate the volume of the hemisphere formed by the bowl and then subtract the volume of the remaining portion when the water is present.
1. Volume of the hemisphere:
The formula to calculate the volume of a hemisphere is (2/3) * π * r³, where r is the radius of the hemisphere.
Using the given radius, we can calculate the volume of the hemisphere as:
(2/3) * π * (30 cm)³ = 60,000π cm³
2. Volume of the remaining portion when the water is present:
To find the volume of the remaining portion, we need to subtract the volume of the water-filled part from the volume of the hemisphere. Since the water depth is given as 15 cm, the remaining portion will be a smaller hemisphere.
The formula to calculate the volume of a hemisphere is the same as before: (2/3) * π * r³, but this time, the radius is reduced.
The radius of the smaller hemisphere can be calculated using the Pythagorean theorem as follows:
radius of smaller hemisphere = √((radius of larger hemisphere)² - (depth of water)²)
radius of smaller hemisphere = √((30 cm)² - (15 cm)²) = √(900 cm² - 225 cm²) = √675 cm ≈ 26 cm
Now, we can calculate the volume of the remaining portion of the hemisphere:
(2/3) * π * (26 cm)³ ≈ 35,373π cm³
3. Volume of water:
To find the volume of water, we subtract the volume of the remaining portion from the volume of the hemisphere:
Volume of water = Volume of hemisphere - Volume of remaining portion
≈ 60,000π cm³ - 35,373π cm³
≈ 24,627π cm³
To find the volume to the nearest cubic centimeter, we can approximate π as 3.14 and calculate the final volume:
Volume of water ≈ 24,627 * 3.14 cm³ ≈ 77,386 cm³
Therefore, the volume of water in the hemispherical bowl, to the nearest cubic centimeter, is approximately 77,386 cm³.
Volume of a sphere = 4/3 * pi * r^3
You volume of water is half of that.