Use the relationship among the formulas for the volumes of cones cylinders and spheres to find the following problem the volume of a cylinder is 36 CM what is the volume of a sphere if it's radius is the same as the cylinders and the height of the cylinder is equal to the Spheres diameter
To find the volume of the sphere, we need to first determine the radius of the cylinder. We can use the formula for the volume of a cylinder:
Volume of cylinder = π * radius^2 * height
From the given information, we know that the volume of the cylinder is 36 cm^3. Let's substitute this into the formula:
36 = π * radius^2 * height
Since the height of the cylinder is equal to the diameter of the sphere, we can express the height of the cylinder as 2 * radius (because diameter = 2 * radius). So, our equation becomes:
36 = π * radius^2 * 2 * radius
To find the volume of the sphere, we also need to determine the radius of the sphere. However, we don't know the relationships between the volumes of spheres and cylinders directly. Instead, we can use the relationship between the volumes of a sphere and a cone.
The formula for the volume of a cone is:
Volume of cone = (1/3) * π * radius^2 * height
where the height is the slant height of the cone. In a cone that perfectly fits inside a sphere, the slant height is equal to the radius of the sphere.
Now, let's determine the relationship between the volume of the cone and the volume of the cylinder. Since we know that the height of the cylinder is equal to the diameter of the sphere, which is twice the radius of the sphere, we have:
2 * radius = height = radius of cylinder
So, the radius of the sphere is equal to the radius of the cylinder.
Now, let's compare the volumes of the cone and the cylinder:
(1/3) * π * radius^2 * 2 * radius = π * radius^2 * height
After canceling out the common terms, we get:
(1/3) * 2 = 1
So, the volume of the cone is one-third of the volume of the cylinder.
Now, we can determine the volume of the cone by multiplying the volume of the cylinder by one-third:
Volume of cone = (1/3) * Volume of cylinder = (1/3) * 36 = 12 cm^3
Since the volume of the cone is equal to the volume of the sphere when the cone perfectly fits inside the sphere, the volume of the sphere is:
Volume of sphere = Volume of cone = 12 cm^3
Therefore, the volume of the sphere with the same radius as the cylinder and height equal to the diameter of the sphere is 12 cm^3.