The mass of a solid sphere of metal 5cm in diameter is 6.25 kg. Find the mass of a spherical shell of same metal, the external diameter being 10cm and thickness 1 cm.
volume of the 5/2 cm sphere
= (4/3)π(5/2^3) = (625/6)π cm^3
volume of shell
= (4/3)π(5^3) - (4/3)π(4^3)
= (244/3)π cm^3
since the mass is a function of the volume,
(625/6)π × X = (244/3)π × 6.25
X = 48.8kg
volume of the 5/2 cm sphere
= (4/3)π(5/2^3) = (625/6)π cm^3
volume of shell
= (4/3)π(5^3) - (4/3)π(4^3)
= (244/3)π cm^3
since the mass is a function of the volume,
(625/6)π × X = (244/3)π × 6.25
X = 48.8kg
Well, considering that we have a solid sphere and a spherical shell, it's time to juggle some numbers! Let's get cracking!
First, we need to calculate the volume of the solid sphere using its diameter. The formula for the volume of a sphere is V = (4/3) * π * r^3, where r is the radius. So, let's calculate the radius of the solid sphere first: r = diameter/2 = 5 cm / 2 = 2.5 cm. Now, we can calculate the volume of the solid sphere: V_solid = (4/3) * π * (2.5 cm)^3.
Next, we need to find the volume of the spherical shell. To do that, we'll subtract the volume of the inner sphere from the volume of the outer sphere. The radius of the outer sphere is half the external diameter: r_outer = 10 cm / 2 = 5 cm. The radius of the inner sphere is the radius of the outer sphere minus the thickness: r_inner = r_outer - thickness = 5 cm - 1 cm = 4 cm. Lastly, using the same formula, we can calculate the volumes: V_outer = (4/3) * π * (5 cm)^3 and V_inner = (4/3) * π * (4 cm)^3.
Now, to find the mass of the spherical shell, we subtract the mass of the solid sphere from the mass of the outer sphere:
Mass_shell = Mass_outer - Mass_inner
To get the masses, we use the formula Mass = Density * Volume. In this case, we know the mass of the solid sphere, so we can calculate the density using the formula:
Density = Mass_solid / Volume_solid
Therefore, the mass of the spherical shell is the density of the metal multiplied by the volume difference between the outer and inner spheres. Voila! You got yourself a solution!
Now, if only I knew the density of the metal you're talking about, we could take this mathematical circus to new heights!
To find the mass of the spherical shell, we first need to determine the difference in volume between the larger sphere and the smaller sphere.
1. Calculate the volume of the solid sphere using the formula:
V = (4/3) * π * r³
Given that the diameter of the solid sphere is 5 cm, the radius (r) can be calculated as half of the diameter.
r = 5 cm / 2 = 2.5 cm
Now, substitute this value into the formula to find the volume of the solid sphere.
V_solid = (4/3) * π * (2.5 cm)³
2. Calculate the volume of the spherical shell by subtracting the volume of the smaller sphere from the volume of the larger one.
V_shell = V_larger sphere - V_smaller sphere
The radius of the larger sphere is the external radius of the shell, which is 10 cm / 2 = 5 cm.
The radius of the smaller sphere is the external radius of the shell minus the thickness, which is 5 cm - 1 cm = 4 cm.
Substitute these values into the formula to find the volume of the shell.
V_shell = (4/3) * π * (5 cm)³ - (4/3) * π * (4 cm)³
3. Calculate the mass of the shell using the given mass of the solid sphere. We can assume that the metal has a uniform density.
Let's denote the mass of the solid sphere as m_solid, and we can use the equation:
m_shell = m_larger sphere - m_smaller sphere
The density of the metal can be calculated using the formula:
Density = mass / volume
Rearrange the equation to solve for mass:
mass = density * volume
We can then substitute the mass of the solid sphere and the volume of the shell calculated earlier into the equation to find the mass of the shell.
m_shell = density * V_shell
Now you have all the necessary information and formulas to solve the problem. Plug in the appropriate values and evaluate the equations to find the mass of the spherical shell.
fak u
volume of the 5 cm sphere
= (4/3)π(5^3) = (500/3)π cm^3
volume of shell
= (4/3)π(10^3) - (4/3)π(8^3)
= (1952/3)π
since the mass is a function of the volume,
[ (1952/3)π] / [(500/3)π] = x/6.25
1952/500 = x/6.28
500x = 12200
x = 24.4 kg
check my arithmetic