Question

8 solid iron sphare with radius 'a cm' each are melted to form a sphare with radius 'b cm'. find the ratio of a:b​

Answer 1

8 solid iron sphere with radius 'a cm' each are melted to form a sphere with radius 'b cm' then the ratio of a:b is 1 : 2

Solution:

Given that 8 solid iron sphere with radius 'a cm' each are melted to form a sphere with radius 'b cm'

Need to find the ratio of a:b

As 8 solid iron sphere with radius 'a cm' each are melted to form a sphere with radius 'b cm'.

For sake of simplicity, let volume of 1 sphere of radius a cm is represented by
`V_a` and volume of 1 sphere of radius b cm is represented by
`V_b`

So volume of 8 solid iron sphere with radius 'a cm' = volume of 1 solid iron sphere with radius 'b cm'

`=>8 *} \mathrm{V}_{\mathrm{a}}=\mathrm{V}_{\mathrm{b}}`

`\frac{\mathrm{V}_{\mathrm{a}}}{\mathrm{V}_{\mathrm{b}}}=(1)/(8)` ---- eqn 1

`\text {Let's calculate } {V}_(a) \text { and } V_(b)`

Formula for volume of sphere is as follows:

`V=(4)/(3) \pi r^(3)`

Where r is radius of the sphere

Substituting r = a cm in the formula of volume of sphere we get

`V_(a)=(4)/(3) \pi r^(3)=(4)/(3) \pi a^(3)`

Substituting r = b cm in the formula of volume of sphere we get

`V_(b)=(4)/(3) \pi r^(3)=(4)/(3) \pi b^(3)`

`\text { Substituting value of } V_(a) \text { and } V_(b) \text { in equation }(1) \text { we get }`

`((4)/(3) \pi a^(3))/((4)/(3) \pi b^(3))=(1)/(8)`

`\begin{array}{l}{=>((4)/(3) \pi a^(3))/((4)/(3) \pi b^(3))=(1)/(8)} \\\\ {=>\left((a)/(b)\right)^(3)=\left((1)/(2)\right)^(3)} \\\\ {=>(a)/(b)=(1)/(2)}\end{array}`

a : b = 1 : 2

Hence the ratio of a:b is 1 : 2