Using the relationship among the formulas for the volume of cone cylinders and spheres to solve the following problem the volume of a cylinder is 36 cm 3 what is the volume of a spirit if it's radius is the same as the cylinders and the height of the cylinder is equal to the spheres demeanor
To solve this problem using the relationship among the formulas for the volume of cone cylinders and spheres, we need to determine the radius and height of the sphere.
Given that the volume of the cylinder is 36 cm^3, we can use the formula for the volume of a cylinder:
V_cylinder = πr^2h
where r is the radius and h is the height.
Since the volume of the cylinder is given as 36 cm^3, we have:
36 = πr^2h
Now, the volume of a sphere is given by the formula:
V_sphere = (4/3)πr^3
To find the volume of the sphere, we need to determine the radius and the height of the cylinder.
Using the relationship between the radius and height of the cylinder, we have:
diameter = 2r = h
Substituting h = 2r into the equation for the volume of the cylinder, we get:
36 = πr^2(2r)
36 = 2πr^3
Dividing both sides by 2π, we have:
r^3 = 18/π
Taking the cube root of both sides, we get:
r = (18/π)^(1/3)
Now that we have the radius of the sphere, we can substitute it into the formula for the volume of the sphere to find the volume:
V_sphere = (4/3)π(r^3)
V_sphere = (4/3)π[(18/π)^(1/3)]^3
Simplifying, we have:
V_sphere = (4/3)π(18/π)
V_sphere = 24 cm^3
Therefore, the volume of the sphere is 24 cm^3.