Question

Given that the volume of a sphere varies as the cube of its radius and its volume is $179.7c{m}^3$ when radius $3.5cm$, find the volume when radius is $1.75cm$

Answer 1

The statement " $x$ varies directly as $y$ ," means that when $y$ increases, $x$ increases by the same factor. In other words, $y$ and $x$ always have the same ratio that is:

$x=ky$

where $k$ is the constant of variation.

Here, it is given that the volume $V$ varies as the cube of its radius $r^3$ if $V=179.7$ cm${}^3$ and $r=3.5$ cm, therefore,

$V=k{r}^3\Rightarrow 179.7=k{\left(3.5\right)}^3\Rightarrow 179.7=k\times 42.875\Rightarrow k=\frac{179.7}{42.875}\Rightarrow k=4.2$

Thus, the general equation is $V=4.2r^3$.

Since $r=1.73$ cm, therefore,

$V=4.2{\left(1.75\right)}^3\Rightarrow V=4.2\times 5.36=22.45$

Hence, the volume of the sphere is $22.45$cm${}^3$when radius is $1.75$ cm.