A spherical container which is 30cm in diameter is 3/4 full of water. The water is emptied into a cylindrical container of diameter 12cm.What is the depth of the water in the cylindrical container.
6^2 * h = 3/4 * 4/3 * 15^3
To find the depth of the water in the cylindrical container, we need to compare the volumes of water in the spherical container and the cylindrical container.
The volume of a sphere is given by the formula V = (4/3) * π * r^3, where r is the radius.
Given that the diameter of the spherical container is 30 cm, the radius is half of that, which is 15 cm.
So, the volume of water in the spherical container is:
V_sph = (4/3) * π * (15 cm)^3 = (4/3) * π * 3375 cm^3 = 4500π cm^3
Now let's calculate the volume of water in the cylindrical container. The formula for the volume of a cylinder is V = π * r^2 * h, where r is the radius and h is the height.
Given that the diameter of the cylindrical container is 12 cm, the radius is half of that, which is 6 cm.
We know that the volume of water in the cylindrical container is 3/4 of the volume in the spherical container, so we can set up the following equation:
V_cyl = (3/4) * 4500π cm^3
Let's solve for the height (h):
(3/4) * 4500π = π * (6 cm)^2 * h
Simplifying the equation:
(3/4) * 4500 = 36 * h
Dividing both sides by 36:
h = (3/4) * 125 = 93.75 cm
Therefore, the depth of the water in the cylindrical container is 93.75 cm.
To find the depth of the water in the cylindrical container, we need to first calculate the volume of the water that was poured into it.
1. Volume of the spherical container:
The volume of a sphere can be calculated using the formula V = (4/3)πr^3, where r is the radius of the sphere. Given that the diameter of the sphere is 30 cm, the radius would be half of that, which is 15 cm.
Plugging in the values, we get V = (4/3) × π × (15 cm)^3 = 4,500π cm^3.
2. Volume of the water in the spherical container:
Since the spherical container is 3/4 full of water, we can calculate its volume by simply multiplying the volume of the spherical container by 3/4:
Volume of water = (3/4) × 4,500π cm^3 = 3,375π cm^3.
3. Volume of the cylindrical container:
The volume of a cylinder can be calculated using the formula V = πr^2h, where r is the radius and h is the height (or depth) of the cylinder. Given that the diameter of the cylindrical container is 12 cm, the radius would be half of that, which is 6 cm.
Plugging in the values, we get V = π × (6 cm)^2 × h = 36πh cm^3.
4. Equating the volumes of the water in the two containers:
Since the water from the spherical container is poured into the cylindrical container, the volume remains the same. Therefore, we can set the volume of the water in the cylindrical container equal to the volume of the water in the spherical container:
36πh cm^3 = 3,375π cm^3.
5. Solving for h (the depth of the water in the cylindrical container):
To find the value of h, we can divide both sides of the equation by 36π:
h = (3,375π cm^3) / (36π) = 93.75 cm.
Hence, the depth of the water in the cylindrical container is 93.75 cm.