two solid spheres have surface area of 5cm^2 and 45cm^2 respectively and mass of the smaller sphere is 2kg. find the mass of the larger sphere

Question

geometry of o levels

Answer 1

Mass and volume have a proportional relationship as can be seen in the equation m/v=d (mass/volume=density).

 

Since the two objects are made of the same material, we do not need to worry about density.

 

Since they are proportionally related, we can represent it as m₁/v₁=m₂/v₂.

 

Volume and surface area are exponentially related. For a square, when you double the lengths of the sides, the area quadruples, or 2², and when you double the lengths of all of the sides of a cube, the volume is 8 times larger, or 2³. If we want to know the rate of change in v we must first find the rate of change in the length.

 

The change in area equals the square in the change of length: a=l²

 

The change in volume equals the cube in the change of length: v=l³

 

Solving for length in both equations gives us l=a^(1/2) and l=v^(1/3)

 

Now we can substitute v in for the first euation, giving us v^(1/3)=a^(1/2)

 

Next we cube each side to get v by itself: v=a^(3/2)

 

So the rate of change in volume equals the square root of the rate of change in the surface area cubed.

 

The rate of change of area is 45/5=9

 

So v=9^(3/2)

 

9^(1/2)=3 and 3³=27 so 9^(3/2)=27

 

So the rate of change in volume is 27: v₁*27=v₂

 

This means the rate of change in mass is 27 also: m₁*27=m₂

 

Since m1 is the smaller sphere we now have 2*27=m₂

 

2*27=54

 

The larger sphere's mass is 54kg.