Aluminum floater: A spherical shell with an outer radius R=2 cm is made of aluminum (mass density of Al is 2.7 g/cm3). How thick is the shell if the average density of the spherical object is exactly 0.5 g/cm3 ? Assume there is a vacuum in the hollow interior of the shell.

Would be awesome if someone could get me going on this or have guided steps how to solve. my brain doesnt want to function and im stuck. THANKS

The volume of the shell and its interior is

v = 4/3 πr^3
= 4/3 π * 2^3
= 33.510 cm^3

That means that the mass (volume*density) is

m = 33.510cm^3 * 0.5g/cm^3 = 16.755g

So, if the interior radius of the shell is r, then the volume of the actual shell material is

v = 4/3 π(2^3-r^3) = 33.510-4.189r^3

So, since volume = mass/density, (using the actual density of Al),

33.510-4.189r^3 = 16.755/2.7
r = 1.868

So, the thickness is 0.132 cm

No problem! I'll be glad to help you out.

To find the thickness of the spherical shell, we can start by considering the concept of density and the mass of the shell.

Step 1: Calculate the mass of the spherical shell
- The mass of an object can be determined by multiplying its volume by its density.
- The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius of the sphere.
- In this case, since we have a spherical shell, the volume would be the difference between the volume of the outer sphere and the volume of the hollow inner sphere.
- The mass of the shell can then be calculated using the mass density and the obtained volume.

Step 2: Calculate the volume of the shell
- Given that the outer radius (R) is 2 cm and assuming that the inner radius is r (the thickness of the shell), we can write the equation for the volume of the shell as follows:
V_shell = (4/3)πR³ - (4/3)πr³

Step 3: Calculate the mass using the density and volume
- The mass of the shell can be given by the equation:
mass = density * volume

Step 4: Equate the average density with the mass and solve for the thickness
- The average density is given as 0.5 g/cm³.
- Since the total mass is distributed over the volume of the entire sphere, the average density can be calculated by dividing the total mass by the total volume of the sphere.
- Equate the average density with the mass and solve for the thickness (r).

I hope this guide helps you solve the problem. Let me know if you need any further assistance!