Question

as a snowball its mass m decreases at a rate proportional to its surface area .The surface area of a snowball is directly proportional to m2/3

Answer 1

The initial condition is mathematically given as  t=0 and m=125 in the equation.

What is the initial condition?

The given problem tells us that

$&\frac{d m}{d t} \infty m^{2 / 3}$

because  mass m decreases at a rate proportional to its surface

the area so the constant of proportionality is negative

therefore we will take as a constant proportionality

Rewrite the given proportionality equation

$&\frac{d m}{d t} \infty m^{2 / 3}$

$\frac{d m}{d t}=-k^2 m^{2 / 3}$

$\frac{d m}{m^{2 / 3}}=-k^2 d t$

$\int m^{-2 / 3} d m=-k^2 \int d t \quad \text$

$\frac{m^{-2 / 3+1}}{1-\frac{2}{3}}=-k^2 t+C_1 \quad\left$

$C_1=\text { intregal constant and } \int x^n d x=\frac{x^{n+1}}{n+1}\right]$

$&\Rightarrow 3 m^{1 / 3}=-k^2 t+C_1\\\\&\Rightarrow m^{1 / 3}=-\frac{k^2 t}{3}+\frac{C_1}{3}\\\\&\Rightarrow m^{1 / 3}=-\frac{k^2 t}{3}+C \quad \ldots$

$C=\frac{C_1}{3}$

Now if at

t=0

the mass  m=125

$(125)^{1 / 3}=-\frac{k^2 \times 0}{3}+C$

$&\Rightarrow\left(5^3\right)^{1 / 3}=C$

C=5

Substitute }

$C=5 \text { in eq.(1) }$

$&m^{1 / 3}=-\frac{k^2 t}{3}+5\\\\&\Rightarrow m=\left(5-k^2 t / 3\right)^3$

In conclusion, the initial condition should be  t=0 and m=125

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