Eight solid iron spheres are dropped into a cylindrical can of diameter 24cm containing water. If the water level rises by 2cm, calculate the radius of the spheres
To calculate the radius of the spheres, we need to use the information given.
First, let's find the volume of water displaced by the spheres:
Volume of water displaced = π * r^2 * h
Given that the diameter of the can is 24 cm and the water level rises by 2 cm, the height of the displaced water is 2 cm.
24 cm (diameter) = 2 * r (radius)
r = 24 cm / 2
r = 12 cm
Now, let's substitute the values into the formula:
Volume of water displaced = π * (12 cm)^2 * 2 cm
Volume of water displaced = 288π cm^3
Since each sphere has the same volume, we can divide the total volume of water displaced by the number of spheres (8) to find the volume of one sphere:
Volume of one sphere = 288π cm^3 / 8
Volume of one sphere = 36π cm^3
The formula for the volume of a sphere is:
Volume of a sphere = (4/3) * π * r^3
Now, let's equate the volume of one sphere to the formula and solve for the radius:
36π cm^3 = (4/3) * π * r^3
To find the radius, we can cross-multiply and simplify:
3 * 36π cm^3 = 4 *π * r^3
108π cm^3 = 4π * r^3
Canceling out π on both sides:
108 cm^3 = 4 * r^3
Divide both sides by 4:
27 cm^3 = r^3
Take the cube root of both sides:
r = ∛27 cm
r = 3 cm
So, the radius of the spheres is 3 cm.
To find the radius of the spheres, we can use the concept of buoyancy. When the spheres are dropped into the water, they displace some of the water, causing the water level to rise.
Let's first calculate the volume of water displaced by the spheres. We know that the water level rises by 2 cm, which is equal to the height of the displaced water.
The volume of a cylinder is given by the formula V = πr^2h, where "r" is the radius of the cylinder and "h" is the height. In this case, the height is equal to 2 cm, and the radius is half of the diameter, which is 24 cm divided by 2.
So, the volume of water displaced by the spheres is:
V = π(24/2)^2(2)
V = π(12)^2(2)
V = 144π cm^3
Now, since the spheres are solid, the volume of each sphere is given by V = (4/3)πr^3, where "r" is the radius of the sphere.
Since there are eight spheres, the total volume of the spheres is 8 times the volume of one sphere:
8(4/3)πr^3 = 144π
Simplifying, we have:
32πr^3 = 144π
Divide both sides of the equation by 32π:
r^3 = 144π / 32π
r^3 = 4.5
Now, take the cube root of both sides to find the radius "r":
r = ∛(4.5)
r ≈ 1.71 cm
Therefore, the radius of the spheres is approximately 1.71 cm.
πr^2 h = 288π is the volume of water displaced.
So each sphere has a volume of 36π
To find r,
4/3 πr^3 = 36π