Use the relationship among the formulas for the volumes of cone cylinders and spheres to solve the following problem. The volume of a cylinder is 36 cm³ what is the volume of a sphere if it’s a radius is the same as the cylinders and the height of the cylinder is equal to the spheres diameter.
Given that the volume of the cylinder is 36 cm³, we can use the formula for the volume of a cylinder to relate it to the radius and height:
Volume of a cylinder = π * radius² * height
36 = π * radius² * height
Next, since the height of the cylinder is equal to the diameter of the sphere, we can write:
height = 2 * radius (since the diameter of a sphere is twice its radius)
Substituting this value into the previous equation, we get:
36 = π * radius² * 2 * radius
36 = 2π * radius³
Now, we can solve for the radius by dividing both sides of the equation by 2π:
radius³ = 36 / (2π)
radius³ ≈ 5.73
Taking the cube root of both sides, we find:
radius ≈ 1.89
Finally, we can use the formula for the volume of a sphere to calculate the volume of the sphere with the same radius as the cylinder:
Volume of a sphere = 4/3 * π * radius³
Volume of the sphere ≈ 4/3 * π * (1.89)³ ≈ 22.43 cm³
Therefore, the volume of the sphere is approximately 22.43 cm³.