If the weight of a spherical shell is 7/8th of what it would be if it were a solid shell, find the ratio of the inner and outer radii of the shell.
To find the ratio of the inner and outer radii of the spherical shell, we can use the concept of volume.
Let's assume the outer radius of the shell is R and the inner radius is r. We want to find the ratio r/R.
The weight of a solid spherical shell is directly proportional to the volume of the shell. The volume of a spherical shell is given by the difference in volume between the outer and inner spheres:
Volume of the spherical shell = (4/3)πR^3 - (4/3)πr^3
Now, let's consider the weight of the solid shell Ws and the weight of the spherical shell W.
According to the given information, the weight of the spherical shell is 7/8th of what it would be if it were a solid shell:
W = (7/8)Ws
Since weight is directly proportional to volume, we can write:
W = (7/8)Ws
=> Volume of the spherical shell = (7/8) Volume of the solid shell
Now, equating the volume expressions:
(4/3)πR^3 - (4/3)πr^3 = (7/8)((4/3)πR^3)
Simplifying the equation:
(4/3)πR^3 - (4/3)πr^3 = (28/24)πR^3
=> R^3 - r^3 = (21/24)R^3
=> r^3 = (3/24)R^3
=> r^3 = (1/8)R^3
Taking the cube root of both sides:
r = (1/2)R
Therefore, the ratio of the inner radius to the outer radius of the shell is:
r/R = (1/2)R/R = 1/2
Hence, the ratio of the inner and outer radii of the shell is 1:2.
1:2
You give weight data, but ask for distance information.
What is the shell made of?