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 radius when the rate of melting is $1$ gram per second; give the answer correct to $2$ decimal places

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 $M=1$ gm, therefore,

$M{r}^2=240\Rightarrow 1\times r^2=240\Rightarrow r^2=240\Rightarrow r=\sqrt{240}=15.49$

Hence, when the rate of melting is $1$ gm/sec, the radius is $15.49$ cm.