a hollow sphere was practically fill with 300cm³ of water. if a metal stone of sky is emerge so as to fill the shape with water calculate its internal radius and and the surface area of the sphere (π=3.14)
To find the internal radius and surface area of the sphere, we need to use the concept of volume.
Given: A hollow sphere is practically filled with 300 cm³ of water.
The volume of a hollow sphere is given by the formula:
Volume = (4/3) * π * (R³ - r³), where R is the external radius and r is the internal radius.
We can rewrite the formula as:
300 = (4/3) * 3.14 * (R³ - r³)
Dividing both sides by (4/3) * 3.14, we have:
300 / ((4/3) * 3.14) = R³ - r³
225.47 = R³ - r³
We also know that the ratio of the volume of water (300 cm³) to the total volume (4/3 * π * R³ - 4/3 * π * r³) is equal to the ratio of the volume of water to the volume of the solid sphere (4/3 * π * R³):
300 / (4/3 * π * R³ - 4/3 * π * r³) = 300 / (4/3 * π * R³)
300 / (4/3 * π * R³) = 300 / (4/3 * π * r³)
225.47 = R³ / r³
Taking the cube root of both sides, we get:
∛225.47 = R / r
Now, let's solve the equations to find the internal radius (r) and external radius (R).
1. ∛225.47 = R / r
Simplifying, we find: R ≈ 5.73r
2. 225.47 = R³ - r³
Substituting R ≈ 5.73r, we have:
225.47 = (5.73r)³ - r³
225.47 = 186.53r³
Dividing both sides by 186.53, we find:
r³ ≈ 1.21
Taking the cube root of both sides, we get:
r ≈ 1.06 cm
Now, to calculate the surface area of the sphere, we use the formula:
Surface area = 4πR²
Substituting R ≈ 5.73r ≈ 5.73 * 1.06 ≈ 6.07 cm, we find:
Surface area = 4 * 3.14 * 6.07²
Surface area ≈ 463.84 cm²
Therefore, the internal radius is approximately 1.06 cm and the surface area of the sphere is approximately 463.84 cm².