Water is poured into a right circular cone with a height of 10 cm and diameter of 8 cm until it is three-fourths full. All this water is then poured into an empty cylindrical container with a radius of 5 cm and height of 9 cm. What is the height of the water in the cylinder?

Ignoring the factor of pi/3, we have

3/4 * 4^2 * 10 = (5/9 h)^2 * h

Now just solve for h

To find the height of the water in the cylinder, we need to understand the relationship between the volumes of the filled cone and the cylinder.

First, let's calculate the volume of the cone. The formula for the volume of a cone is V = (1/3)πr^2h, where r is the radius and h is the height.

Given that the height of the cone is 10 cm, the radius can be determined by dividing the diameter by 2, since the diameter is given as 8 cm. Thus, the radius is 8/2 = 4 cm.

The volume of the cone when it is filled to three-fourths full is given by multiplying the volume of the entire cone by 3/4:
V_cone = (3/4) * (1/3) * π * (4^2) * 10
= (3/4) * (1/3) * π * 16 * 10
= (3/4) * (π * 16 * 10) / 3
= (3/4) * (5.24 * 16 * 10)
= (3/4) * (83.84 * 10)
= (3/4) * 838.4
= 628.8 cm^3 (approximately)

Next, let's calculate the volume of the cylinder. The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.

Given that the radius of the cylindrical container is 5 cm and the height is 9 cm, we can calculate the volume of the entire cylindrical container:
V_cylinder = π * (5^2) * 9
= π * 25 * 9
= 225π cm^3

Since we poured all the water from the cone into the cylindrical container, the volume of water in the cylinder should be equal to the volume of the filled cone.

Therefore, we need to equate:
V_cone = V_cylinder
628.8 = 225π

To solve for π, divide both sides of the equation by 225:
π = 628.8 / 225
≈ 2.8

Now, we can solve for the height of water in the cylinder using the volume of water in the cylinder and the radius of the cylinder:
V_cylinder = π * r^2 * h
628.8 = 2.8 * 5^2 * h
628.8 = 2.8 * 25 * h
h = 628.8 / (2.8 * 25)
h ≈ 8.96 cm

Therefore, the height of the water in the cylindrical container is approximately 8.96 cm.