Three sphers of diameters 3,4,6 inches respectively are formed into a single sphere.find its diameter supposing that the volume of sphere is proportional to the cube of its diameter.
Answers
We know volume of a cube = 4/3 x pi x r^3 where r is the radius.
Now, acc to question the diameters of the three spheres are 3,4 and 6 inches respectively. So their radius will be 1.5,2 and 3 inches respectively.
Therefore, sum of their volume = 4/3 x 22/7 x ( (1.5^3) + (2^3) + (3^3)) cube inches. [I have taken 4/3 x 22/7 as common from the three while adding to minimise the calculation]
Now, let us assume that the radius of the final sphere is R.
so, it's voume wil be 4/3 x 22/7 x R^3.
and it must be equal to sum of volume of the three small spheres.
Therefore we can equate:
4/3 x 22/7 x ( (1.5^3) + (2^3) + (3^3)) = 4/3 x 22/7 x R^3
from here we can easily evaluate the value of R.
R = cube root of ( (1.5^3) + (2^3) + (3^3)) = 3.375 + 8 + 27 = cube root of 38.375 inches. = 3.372 inches
Thus it's diameter will be twice the radius = 6.74 inches. (Ans)
Answer:
Down vote
Let V1=αD31 be the volume of the first sphere, α being the proportional factor between volume and the cube of the diameter of the sphere.
Define V2 and V3 accordingly.
Then the last shpere whose volume is the sum of the other three has a volume V4
V4=V1+V2+V3
αD34=αD31+αD32+αD33
D4=(D31+D32+D33)1/3
You replace D1=2cm, D2=3cm, D3=4cm and you get your result in cm.