Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter

Question

Three spheres of diameters 2,3&4 cm's respectively formed into a single sphere.Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter

Answer 1

Let $V_1=\alpha D_1^{3}$ be the volume of the first sphere, $\alpha$ being the proportional factor between volume and the cube of the diameter of the sphere.

Define $V_2$ and $V_3$ accordingly.

Then the last shpere whose volume is the sum of the other three has a volume $V_4$

$V_4=V_1+V_2+V_3$

$\alpha D_4^3=\alpha D_1^3 + \alpha D_2^3 + \alpha D_3^3$

$D_4=(D_1^3 + D_2^3 + D_3^3)^{1/3}$

You replace $D_1=2cm$, $D_2=3cm$, $D_3=4cm$ and you get your result in $cm$.

Answer 2

Volume of a sphere: $$V = kd^3$$ So total volume: $$V = k(2^3+3^3+4^3) = k(8+27+64) = k(99)$$ So, the diameter of the new sphere would be $\sqrt[3]{99}$

Answer 3

The shape is irrelevant. For similar shapes, volume varies as the cube of a linear dimension.

So we want $x$ such that

$$\begin{align} x^3 &= 2^3 + 3^3 + 4^3 \\ &= 8 + 27 + 64 \\ &= 99 \end{align}$$

So

$$\begin{align} x &= 99^ {1 \over 3} \\ &\approx 4.626065009182741 \end{align}$$