Use a relationship among the formulas for the volumes of cone cylinders and spheres to solve the following problem the volume of the cylinder is 36 cm cubed what is the volume of the sphere of the radius is the same as the cylinders and the height of the cylinders equal to the spheres diameter
18 cm cubed
54cm cubed
12 cm cubed
24 cm cubed
The volume of a cylinder is given by the formula V_cylinder = πr^2h, where r is the radius and h is the height.
The volume of a cone is given by the formula V_cone = (1/3)πr^2h, where r is the radius and h is the height.
The volume of a sphere is given by the formula V_sphere = (4/3)πr^3, where r is the radius.
In this problem, we are given that the volume of the cylinder is 36 cm^3.
Since the height of the cylinder is equal to the diameter of the sphere, we can say that h = 2r (where r is the radius of both the cylinder and the sphere).
Substituting this value of h = 2r into the formula for the volume of the cylinder:
36 = πr^2(2r)
36 = 2πr^3
r^3 = 18/π
r ≈ 1.53 cm
Now we can find the volume of the sphere using the formula:
V_sphere = (4/3)π(1.53)^3
V_sphere ≈ 29.12 cm^3
Therefore, the volume of the sphere is approximately 29.12 cm^3.
Wrong it's either 18,54 ,12 or 24
Apologies for the confusion. Let's solve the problem again.
Given that the volume of the cylinder is 36 cm^3.
Using the formula for the volume of a cylinder:
V_cylinder = πr^2h
We can rewrite this equation in terms of r and h:
36 = πr^2h
Since the height of the cylinder is equal to the diameter of the sphere, we can express h in terms of the radius as h = 2r.
Substituting this value into the equation:
36 = πr^2(2r)
36 = 2πr^3
Dividing both sides by 2π:
18 = πr^3
Now, let's find the volume of the sphere using the formula:
V_sphere = (4/3)πr^3
Substituting the value of πr^3 from the previous equation:
V_sphere = (4/3)π(18)
Simplifying:
V_sphere = 24π
Since we're given options in terms of centimeters cubed, we can approximate π as 3.14:
V_sphere ≈ 24(3.14)
V_sphere ≈ 75.36 cm^3
Therefore, the volume of the sphere is approximately 75.36 cm^3, so the answer is 75 cm^3.