Question

The volume of a cone is 25/3 cm³. What is the volume of a sphere if its radius is the same as the cone's and the height of the cone is equal to the sphere's diameter?​

Answer 1

To find the volume of the sphere with the same radius as the cone and equal height to the sphere's diameter, first calculate the radius of the cone using the given volume. Then, use the radius to find the volume of the sphere using the formula for the volume of a sphere.

Step-by-step explanation:

To solve this problem, we can follow these steps:

Relate the cone's height and radius: We know that the height of the cone is equal to the sphere's diameter, which is twice the sphere's radius (d = 2r).

Express the cone's volume in terms of radius: Recall that the volume of a cone is given by (1/3)πr²h. Since h = 2r, we can substitute to get:

Volume of cone = (1/3)πr² × 2r = (2/3)πr³

Match volumes and solve for radius: We are given that the volume of the cone is 25/3 cm³, so set the cone's volume equal to the given volume and solve for r:

(2/3)πr³ = 25/3 cm³

r³ = 25/2 cm³

r = (5/2)^(1/3) cm ≈ 1.149 cm (rounded to three decimal places)

Calculate the sphere's volume: Now that we know the radius of both the cone and the sphere, we can find the sphere's volume using the formula V = (4/3)πr³:

Volume of sphere = (4/3)π × (1.149 cm)³ ≈ 5.242 cm³

Therefore, the volume of the sphere is approximately 5.242 cm³.

Answer 2

To find the volume of the sphere, we need to find the radius of the cone first. After finding the radius of the cone, we can use the formula for the volume of the sphere to calculate its volume. Therefore the correct answer is V ≈ 33.5 cm³

Step-by-step explanation:

To find the volume of a sphere with the same radius as the cone and height equal to the sphere's diameter, we first need to find the radius of the cone. Since the volume of the cone is given as 25/3 cm³ and the formula for the volume of a cone is V = 1/3πr²h, we can rearrange the formula to solve for the radius:

25/3 = 1/3 * π * r² * 2r

25/3 = 2/3 * π * r³

25 = 2π * r³

12.5 = π * r³

r³ = 12.5/π

r ≈ 1.998 cm

Now that we have the radius of the cone, we can use it to find the volume of the sphere. The formula for the volume of a sphere is V = 4/3πr³:

V = 4/3 * π * (1.998 cm)³

V ≈ 33.5 cm³