Hollow sphere

Steel hollow sphere floats on the water plunged into half its volume. Determine the outer radius of the sphere and wall thickness, if you know that the weight of the sphere is 0.5 kg and density of steel is 7850 kg/m³

Result

r =  6.2 cm
h =  0.135 cm

Solution:

$\ \\ \rho_1 = 7850 kg/m^3 \ \\ \rho_2 = 1000 kg/m^3 \ \\ \ \\ \dfrac12 \rho_2 \dfrac43 \pi r^3 = 0.5 \ kg \ \\ r = \sqrt[3]{\dfrac34 \cdot 2\cdot 0.5 / \pi / 1000 } \ \\ r = 0.062035 \ m = 6.2 \ \text{cm} \ \\ \ \\ \rho_1 \dfrac43 \pi r^3 - \rho_1 \dfrac43 \pi (r-h)^3 = 0.5 \ kg \ \\ \ \\ r^3-(r-h)^3 = \dfrac{0.5 \cdot 3}{ \rho_1 4 \pi } \ \\ (r-h)^3 = r^3 - \dfrac{0.5 \cdot 3}{ \rho_1 4 \pi } \ \\ r -h = \sqrt[3]{ r^3 - \dfrac{0.5 \cdot 3}{ \rho_1 4 \pi } } \ \\ h = r - \sqrt[3]{ r^3 - \dfrac{0.5 \cdot 3}{ \rho_1 4 \pi } } \ \\$

$h = 0.135 \ \text{cm}$

Try another example

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Showing 0 comments:

Be the first to comment!

Next similar math problems:

1. Hollow sphere
   Calculate the weight of a hollow tungsten sphere (density 19.3 g/cm³), if the inner diameter is 14 cm and wall thickness is 3 mm.
2. Hollow sphere
   The volume of the hollow ball is 3432 cm³. What is its internal radius when the wall thickness is 3 cm?
3. Wall thickness
   The hollow metal ball has an outside diameter of 40 cm. Determine the wall thickness if the weight is 25 kg and the metal density is 8.45 g/cm³.
4. Gasholder
   The gasholder has spherical shape with a diameter 20 m. How many m³ can hold in?
5. Shots
   5500 lead shots with diameter 4 mm is decanted into a ball. What is it diameter?
6. Sphere growth
   How many times grow volume of sphere if diameter rises 10×?
7. Volume of ball
   Find the volume of a volleyball that has a radius of 4 1/2 decimeters. Use 22/7 for π
8. Plasticine ball
   Plasticine balls have radius r₁=85 cm, r₂=60 mm, r₃=59 cm, r₄=86 cm, r₅=20 cm, r₆=76 mm, r₇=81 mm, r₈=25 mm, r₉=19 mm, r₁₀=14 cm. For these balls
9. Sphere fall
   How many percent fall volume of sphere if diameter fall 10×?
10. Holidays - on pool
    Children's tickets to the swimming pool stands x € for an adult is € 2 more expensive. There was m children in the swimming pool and adults three times less. How many euros make treasurer for pool entry?
11. Water in vegetables
    Tomatoes in the store contain 99% water. After being transported to the shop, they were slightly dried and contained only 98% of water. How many kgs of tomatoes are in the store if there were 300 kg in stock?`
12. Two boxes-cubes
    Two boxes cube with edges a=38 cm and b = 81 cm is to be replaced by one cube-shaped box (same overall volume). How long will be its edge?
13. The volume 2
    The volume of a cube is 27 cubic meters. Find the height of the cube.
14. Cuboid height
    What is the height of the cuboid if the edges of its base are 15 cm and 4 cm long and its volume is 420 cm cubic?
15. Theorem prove
    We want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started?
16. The swing
    To swing the two girls. Aneta weight 45 kg and Simon 35 kg weight. How far should sit Simon from the middle of swing so it is balanced, if we know that Aneta is sitting at distance 1,5m? How far are girls sitting apart?