Question

The rate of change of the volume of a snowball that is melting is proportional to the surface area of the snowball.

Upper Level Math
ANSWERED
asked 2021-01-02

The rate of change of the volume of a snowball that is melting is proportional to the surface area of the snowball. Suppose the snowball is perfectly spherical. Then the volume (in centimeters cubed) of a ball of radius r centimeters is \(\displaystyle\frac{{4}}{{3}}\pi{r}^{{3}}\). The surface area is \(\displaystyle{4}\pi{r}^{{2}}\).Set up the differential equation for how r is changing. Then, suppose that at time \(t = 0\) minutes, the radius is 10 centimeters. After 5 minutes, the radius is 8 centimeters. At what time t will the snowball be completely melted.

Expert Answers (1)

2021-01-03

Step 1
Given:
The rate of change of the volume of a snowball that is melting is proportional to the surface area of the snowball is perfectly spherical
Then the volume (in centimetres cubed) of a ball of radius r centimetres is
\(\displaystyle{v}=\frac{{4}}{{3}}\pi{r}^{{3}}\)
And the surface area is
\(\displaystyle{s}={4}\pi{r}^{{2}}\)
Set up the differential equation for how r is changing.
Then, suppose that at time t=0 minutes,the radius is 10 centimetres.After 5 minutes,the radius is 8 centimetres
Step 2
To find: At the what time t will be snowball be completely melted?
From the given conditions ,
\(\displaystyle\frac{{{d}{V}}}{{{\left.{d}{t}\right.}}}{\left(\alpha{S}\right)}\)
\(\displaystyle\frac{{{d}{V}}}{{{\left.{d}{t}\right.}}}{\left(\lambda{S}\right)}\)......(1)
\(\displaystyle{V}=\frac{{4}}{{3}}\pi{r}^{{3}}\)
\(\displaystyle\frac{{{d}{V}}}{{{\left.{d}{t}\right.}}}=\frac{{4}}{{3}}\pi{3}{r}^{{2}}\frac{{{d}{r}}}{{{\left.{d}{t}\right.}}}\)
By putting this value in (1)
The equation must be,
\(\displaystyle\frac{{4}}{{3}}\pi{3}{r}^{{2}}\frac{{{d}{r}}}{{{\left.{d}{t}\right.}}}=\lambda{4}\pi{r}^{{2}}\)
\(\displaystyle\frac{{{d}{r}}}{{{\left.{d}{t}\right.}}}=\lambda\)
\(\displaystyle{r}=\lambda{t}+{c}\)
Now, \(t = 0\) and \(r = 10\)
so,
\(\displaystyle{r}=\lambda{t}+{c}\)
\(\displaystyle{10}=\lambda{\left({0}\right)}+{c}\)
c=10 and here
\(\displaystyle{r}=\lambda{t}+{10}\)......(2)
After the 5 minutes, \(t = 5\) and \(r = 8\)
\(\displaystyle{8}=\lambda{\left({5}\right)}+{10}\)
\(\displaystyle{5}\lambda=-{2}\)
\(\displaystyle\lambda=-\frac{{2}}{{5}}\)
Now equation (2)becomes,
\(\displaystyle{r}=-\frac{{2}}{{5}}{t}+{10}\)......(3)
This shows the differential equation for how r is changing.
As the snowball completely melted that means the radius of the snowball is zero.
From this by substituting \(r=0\) in the equation (3) must be,
\(\displaystyle{0}=-\frac{{2}}{{5}}{t}+{10}\)
\(\displaystyle\frac{{2}}{{5}}{t}={10}\)
\(\displaystyle{t}=\frac{{{10}\times{5}}}{{2}}\) \(t=25\) Hence the time required for melting the snowball is 25 minutes.

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