Question

Suppose the rate of melting, $M$ grams per second, of a sphere of ice is inversely proportional to the square of the radius, $rcm$. When $r=20,M=0.6$. Find the rate of melting when $r=40cm$.

Answer 1

The statement " $y$ varies inversely as $x$ means that when $x$ increases, $y$ decreases by the same factor. In other words, the expression $xy$ is constant:

$xy=k$

where $k$ is the constant of variation.

Here, it is given that the rate of melting $M=0.6$ g/sec of a sphere of ice is inversely proportional to the square of radius $r=20$ cm, therefore,

$M{r}^2=k\Rightarrow 0.6\times {\left(20\right)}^2=k\Rightarrow k=0.6\times 400\Rightarrow k=0.6\times 400\Rightarrow k=240$

Thus, the general equation is $M{r}^2=240$.

Since $r=40$ cm, therefore,

$M{r}^2=240\Rightarrow M{\left(40\right)}^2=240\Rightarrow M\times 1600=240\Rightarrow M=\frac{240}{1600}=\frac{3}{20}$

Hence, the rate of melting when $r=40$ cm is $\frac{3}{20}$ gms.