A sphere of radius 4 cm iss carved from a homogenous sphere of radius 8cm and mass 160g. the mass of the smaller sphere is
Since the material is same, the density is also same. The formula for mass of the sphere m =Volume x density
=4/3 (pi) (r)^3 x density
m1= 160g, r1=8cm.
r2=4cm, m2=?
160=4/3 (pi) (8)^3x density-----(1)
m2=4/3 (pi) (4)^3x density-----(2)
now equation (2)/(1)
m2/160= (4)^3/(8)^3
m2=(4/8)^3 x 160
=160/8
=20 gm
simple formula is
m2=(r2/r1)^3 x m1
m2=(4/8)^3*160
=20g
Well, if we consider the larger sphere as a clown's nose and the smaller sphere as a mini clown nose, it's like a clown decided to upgrade his style!
To find the mass of the smaller sphere, we can use the formula for the volume of a sphere: V = (4/3)πr³, where r is the radius.
The volume of the larger sphere would be V₁ = (4/3)π(8 cm)³, and the volume of the smaller sphere would be V₂ = (4/3)π(4 cm)³.
The ratio of the volumes is V₂/V₁ = [(4/3)π(4 cm)³]/[(4/3)π(8 cm)³] = (4 cm/8 cm)³ = (1/2)³ = 1/8.
Since mass is directly proportional to volume, we can say that the mass of the smaller sphere is 1/8th of the mass of the larger sphere.
So, the mass of the smaller sphere would be 1/8 * 160g = 20g.
So, the mass of the smaller sphere is 20 grams. That's one lightweight, fashionable clown nose!
To find the mass of the smaller sphere, we can use the concept of volume and density.
The volume of a sphere can be calculated using the formula:
V = (4/3) * π * r^3
where V is the volume and r is the radius.
The larger sphere has a radius of 8 cm, so its volume is:
V_larger = (4/3) * π * (8^3)
The smaller sphere has a radius of 4 cm, so its volume is:
V_smaller = (4/3) * π * (4^3)
The ratio of the volumes, V_smaller / V_larger, gives us the ratio of the masses of the two spheres since the larger sphere is homogeneous.
mass_smaller / mass_larger = V_smaller / V_larger
mass_smaller / 160g = V_smaller / V_larger
mass_smaller = (V_smaller / V_larger) * 160g
Substituting the values:
mass_smaller = [(4/3) * π * (4^3)] / [(4/3) * π * (8^3)] * 160g
mass_smaller = (4^3 / 8^3) * 160g
mass_smaller = (64/512) * 160g
mass_smaller = 0.125 * 160g
mass_smaller = 20g
Therefore, the mass of the smaller sphere is 20 grams.
To find the mass of the smaller sphere, we need to consider the property of homogeneity.
The property of homogeneity means that the density of the material is constant throughout the object. Therefore, we can assume that the smaller sphere has the same density as the larger sphere.
The volume of a sphere is given by the formula V = (4/3)πr^3, where r is the radius.
For the larger sphere with a radius of 8 cm, the volume can be calculated as:
V_large = (4/3)π(8^3) = (4/3)π(512) = 2144π cm^3
Similarly, for the smaller sphere with a radius of 4 cm, the volume can be calculated as:
V_small = (4/3)π(4^3) = (4/3)π(64) = 256π cm^3
Since the density is assumed to be the same for both spheres, we can write the following equation:
(density_large) * (V_large) = (density_small) * (V_small)
We know the mass of the larger sphere is 160 g. Therefore, we can rearrange the equation to solve for the mass of the smaller sphere:
mass_small = (density_large * V_large) / V_small
To find the density, we divide the mass by the volume: density = mass / volume.
density_large = 160 g / (2144π cm^3)
density_small = mass_small / (256π cm^3)
Now, substitute these values back into the equation:
mass_small = ((160 g) / (2144π cm^3)) * (256π cm^3)
mass_small = (160 g * 256π cm^3) / (2144π cm^3)
mass_small = (160 g * 256) / 2144
mass_small = 1920 g / 2144
mass_small = 0.896 g (rounded to three decimal places)
Therefore, the mass of the smaller sphere is approximately 0.896 grams.