Two spheres of copper, of radii is 1cm and 2cm respectively, are melted into a cylinder of radius 1cm. Find the altitude of the cylinder.
for sphere, v = 4π/3 r^3
for cylinder, v = πr^2h
so, since the radius of the cylinder is 1,
4π/3 (1^3+2^3) = πh
To find the altitude of the cylinder, we need to determine the volume of the two spheres and then find the cylinder's altitude using its volume.
Let's start by finding the volume of each sphere.
The formula to calculate the volume of a sphere is:
V = (4/3) * π * r^3
Where:
V is the volume,
π is approximately 3.14159, and
r is the radius of the sphere.
For the smaller sphere with a radius of 1 cm:
V1 = (4/3) * π * (1^3)
= (4/3) * 3.14159 * 1
≈ 4.18879 cm^3
For the larger sphere with a radius of 2 cm:
V2 = (4/3) * π * (2^3)
= (4/3) * 3.14159 * 8
≈ 33.51032 cm^3
Next, let's find the volume of the entire cylinder by adding the volumes of the two spheres.
V_cylinder = V1 + V2
≈ 4.18879 + 33.51032
≈ 37.69911 cm^3
The formula to calculate the volume of a cylinder is:
V_cylinder = π * r^2 * h
Where:
V_cylinder is the volume of the cylinder,
π is approximately 3.14159,
r is the radius of the cylinder (which is 1 cm), and
h is the altitude (height) of the cylinder.
Now we can rearrange the formula to solve for h:
h = V_cylinder / (π * r^2)
Substituting the known values:
h = 37.69911 / (3.14159 * 1^2)
= 37.69911 / (3.14159 * 1)
≈ 12 cm
Therefore, the altitude of the cylinder is approximately 12 cm.
To find the altitude of the cylinder, we need to analyze the volumes of the spheres and the cylinder.
The volume of a sphere is given by V = (4/3)πr³, where "r" is the radius.
For the smaller sphere with a radius of 1cm, its volume is V1 = (4/3)π(1)³ = (4/3)π cm³.
For the larger sphere with a radius of 2cm, its volume is V2 = (4/3)π(2)³ = (4/3)π(8) = (32/3)π cm³.
Now, let's calculate the volume of the cylinder. The formula for the volume of a cylinder is V = πr²h, where "r" is the radius and "h" is the height or altitude.
Since the radius of the cylinder is also 1cm, its volume is Vc = π(1)²h = πh.
The sum of the volumes of the two spheres is V1 + V2 = (4/3)π + (32/3)π = (36/3)π = 12π cm³.
Since this volume is equal to the volume of the cylinder, we have 12π = πh.
To find "h," we can cancel out the common factor of π from both sides. Thus, h = 12.
Therefore, the altitude of the cylinder is 12 cm.