Question

What is the volume of the larger sphere if the diameters of two spheres are in the ratio of 2:3 and the sum of their volumes is 1260 cu.m?

Answer 1

It is `972` cu.m

The volume formula of spheres is:

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

We have sphere `A` and sphere `B`.

`V_A = (4/3) * pi * (r_A)^3`

`V_B = (4/3) * pi * (r_B)^3`

As we know that `r_A/r_B=2/3`

`3r_A=2r_B`
`r_B=3r_A/2`

Now plug `r_B` to `V_B`

`V_B = (4/3) * pi * (3r_A/2)^3`

`V_B = (4/3) * pi * 27(r_A)^3/8`

`V_B = (9/2) * pi * (r_A)^3`

So we can now see that `V_B` is `(3/4)*(9/2)` times bigger than `V_A`

So we can simplify things now:

`V_A = k`
`V_B = (27/8)k`

Also we know `V_A + V_B = 1260`

`k+(27k)/8 = 1260`

`(8k + 27k)/8 = 1260`

`8k + 27k = 1260*8`
`35k = 10080`
`k = 288`

`k` was the volume of `A` and the total volume was `1260`. So the larger sphere's volume is `1260-288=972`