A sphere with a diameter of 24 inches is reshaped into a cone with a radius of 32 inches. If all the material is used, what is the height of the cone?
sphere: radius = s
cone: radius c, height h
4π/3 s^3 = π/3 c^2 h
4s^3 = c^2 h
4(12^3) = (32^2)h
h = 4*4^3*3^3/2^10 = 3^3/2^2 = 27/4 = 6.75
To find the height of the cone, we can start by finding the volume of the sphere and then using that value to find the height of the cone.
Step 1: Calculate the volume of the sphere
The volume of a sphere can be calculated using the formula:
V = (4/3) * π * r^3
where V is the volume, π is a constant (approximately 3.14), and r is the radius.
Given that the diameter of the sphere is 24 inches, we can find the radius by dividing the diameter by 2:
r = diameter / 2 = 24 inches / 2 = 12 inches
Substituting the value of the radius into the volume formula:
V = (4/3) * π * (12 inches)^3
V ≈ 4.19 * (12 inches)^3
V ≈ 4.19 * 1728 inches^3
V ≈ 7225.6 inches^3
Step 2: Calculate the height of the cone
The volume of a cone can be calculated using the formula:
V = (1/3) * π * r^2 * h
where V is the volume, π is a constant (approximately 3.14), r is the radius, and h is the height.
Given that the radius of the cone is 32 inches, we can substitute the volume of the sphere (7225.6 inches^3) into the volume formula to solve for the height:
7225.6 inches^3 = (1/3) * π * (32 inches)^2 * h
Rearranging the formula to solve for h:
h = (7225.6 inches^3) * 3 / (π * (32 inches)^2)
h ≈ 7386 inches^3 / 3219.92 inches^2
h ≈ 2.29 inches
Therefore, the height of the cone is approximately 2.29 inches.
To find the height of the cone, we can start by calculating the volume of both the sphere and the cone and equating them.
The volume of a sphere can be calculated using the formula: V_sphere = (4/3) * π * r^3, where r is the radius of the sphere.
Given the diameter of the sphere is 24 inches, the radius (r) would be half of the diameter, so r = 24 / 2 = 12 inches.
Plugging this value into the formula gives us: V_sphere = (4/3) * π * (12 inches)^3.
Next, let's calculate the volume of the cone. The volume of a cone can be calculated using the formula: V_cone = (1/3) * π * r^2 * h, where r is the radius of the cone, and h is the height of the cone.
Given that the radius of the cone is 32 inches, the formula becomes: V_cone = (1/3) * π * (32 inches)^2 * h.
Since all the material is used, the volume of the sphere is equal to the volume of the cone: V_sphere = V_cone.
Equating the two formulas, we have:
(4/3) * π * (12 inches)^3 = (1/3) * π * (32 inches)^2 * h.
Canceling out the common terms, we get:
(12 inches)^3 = (32 inches)^2 * h.
Now, solve for h by dividing both sides of the equation by (32 inches)^2:
h = (12 inches)^3 / (32 inches)^2.
Calculating this gives us:
h = 1728 cubic inches / 1024 square inches.
Simplifying further, we find:
h = 1.6875 inches.
Therefore, the height of the cone is approximately 1.6875 inches.