Volume of Cones, Cylinders, and Spheres Quick Check%0D%0A3 of 53 of 5 Items%0D%0A%0D%0AQuestion%0D%0AUse the relationship among the formulas for the volumes of cones, cylinders, and spheres to solve the following problem: The volume of a cylinder is 24x−36 cm3. What is the volume of a sphere if its radius is the same as the cylinder’s and the height of the cylinder is equal to the sphere’s diameter? (1 point)%0D%0AResponses%0D%0A%0D%0A12x−18 cm3%0D%0A12 x minus 18 cm cubed%0D%0A%0D%0A8x−12 cm3%0D%0A8 x minus 12 cm cubed%0D%0A%0D%0A16x−24cm3 %0D%0A16 x minus 24 cm cubed%0D%0A%0D%0A36x−54 cm3
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height.
We are given that the volume of the cylinder is 24x - 36 cm^3. Since the height of the cylinder is equal to the sphere's diameter, we can say that the height of the cylinder is 2r (where r is the radius of the sphere).
Substituting these values into the volume formula for the cylinder, we get:
24x - 36 = πr^2(2r)
24x - 36 = 2πr^3
To find the volume of the sphere, we use the volume formula V = (4/3)πr^3. Comparing the equation above to the sphere's volume formula, we see that the volume of the sphere is (4/3) times the volume of the cylinder:
V_sphere = (4/3)(24x - 36)
V_sphere = 32x - 48 + 8x - 12
V_sphere = 40x - 60
Therefore, the volume of the sphere is 40x - 60 cm^3.
So the correct response is:
40x - 60 cm^3.