Two spheres of copper, of radii is 1 cm. and 2 cm. respectively, are melted in to a cylinder of radius 1 cm. Find the altitude of the cylinder.
volume of the 2 spheres
= (4/3)π (1^3 + 2^3)
= (4/3)π(9) = 12π cm^3
volume of cylinder = π r^2 h
12π = π(r^2)(1)
r^2 = 12
r = √12 or 2√3 cm = appr 3.464 cm
4/3? where is it come from?
To find the altitude of the cylinder, we need to use the principle of conservation of volume.
Step 1: Calculate the volume of each sphere.
The formula for the volume of a sphere is V = (4/3) * π * r^3, where r is the radius of the sphere.
For the first sphere with a radius of 1 cm:
V1 = (4/3) * π * (1^3)
V1 = (4/3) * π * 1
V1 = 4/3 * π
V1 = (4/3)π
For the second sphere with a radius of 2 cm:
V2 = (4/3) * π * (2^3)
V2 = (4/3) * π * 8
V2 = 32/3 * π
Step 2: Find the total volume of the two spheres.
V_total = V1 + V2
V_total = (4/3)π + 32/3π
V_total = (36/3)π
V_total = 12π
Step 3: Find the volume of the cylinder.
The formula for the volume of a cylinder is V = π * r^2 * h, where r is the radius and h is the height or altitude.
Since the cylinder has a radius of 1 cm:
V_cylinder = π * (1^2) * h
V_cylinder = π * 1 * h
V_cylinder = πh
Step 4: Equate the volumes of the spheres and the cylinder.
We know that V_total = V_cylinder, so we can write:
12π = πh
Step 5: Solve for the altitude of the cylinder.
Divide both sides of the equation by π:
12 = h
Therefore, the altitude or height of the cylinder is 12 cm.
To find the altitude of the cylinder, we can use the conservation of volume principle. The volume of the two spheres before they are melted can be calculated using the formula for the volume of a sphere:
Volume of a sphere = (4/3) * π * (radius)^3
For the first sphere with a radius of 1 cm, the volume is:
Volume1 = (4/3) * π * (1 cm)^3 = (4/3) * π * 1 cm^3
For the second sphere with a radius of 2 cm, the volume is:
Volume2 = (4/3) * π * (2 cm)^3 = (4/3) * π * 8 cm^3
Now, when these two spheres are melted, they form a cylinder. The volume of a cylinder can be calculated using the formula:
Volume of a cylinder = π * (radius)^2 * altitude
We already know the radius of the cylinder is 1 cm. Let's assume the altitude of the cylinder is "h" cm. Therefore, the volume of the cylinder is:
Volume of the cylinder = π * (1 cm)^2 * h = π * h cm^3
Since the volume of the spheres before melting is equal to the volume of the cylinder after melting, we can equate them:
Volume1 + Volume2 = Volume of the cylinder
(4/3) * π * 1 cm^3 + (4/3) * π * 8 cm^3 = π * h cm^3
(4/3) * π * (1 cm^3 + 8 cm^3) = π * h cm^3
(4/3) * π * 9 cm^3 = π * h cm^3
Canceling out π and simplifying:
(4/3) * 9 cm^3 = h cm^3
12 cm^3 = h cm^3
Therefore, the altitude (height) of the cylinder is 12 cm.