$6$Camphor is a waxy, colourless solid with strong aroma that evaporates through the process of sublimation, if left in the open at room temperature. 2025_CBSE_Mathematics_Set_Region7_Subset_Set2_Q38___p12_4

2025_CBSE_Mathematics_Set_Region7_Subset_Set2_Q38___p12_4

(Cylindrical-shaped Camphor tablets) A cylindrical camphor tablet whose height is equal to its radius ( $\displaystyle \mathbf{r}$ ) evaporates when exposed to air such that the rate of reduction of its volume is proportional to its total surface area. Thus, $\displaystyle \frac{\mathrm{dV}}{\mathrm{dt}}=\mathrm{kS}$ is the differential equation, where V is the volume, S is the surface area and $\displaystyle \mathrm{t}$ is the time in hours. Based upon the above information, answer the following questions :

(i)Write the order and degree of the given differential equation.(ii)Substituting $\displaystyle \mathrm{V}=\pi \mathrm{r}^{3}$ and $\displaystyle \mathrm{S}=2 \pi \mathrm{r}^{2}$, we get the differential equation $\displaystyle \frac{\mathrm{dr}}{\mathrm{dt}}=\frac{2}{3} \mathrm{k}$. Solve it, given that $\displaystyle \mathrm{r}(0)=5 \mathrm{~mm}$.(iii)(a) If it is given that $\displaystyle \mathrm{r}=3 \mathrm{~mm}$ when $\displaystyle \mathrm{t}=1$ hour, find the value of k . Hence, find t for $\displaystyle \mathrm{r}=0 \mathrm{~mm}$.

OR

(iii) (b) If it is given that $\displaystyle \mathrm{r}=1 \mathrm{~mm}$ when $\displaystyle \mathrm{t}=1$ hour, find the value of k . Hence, find t for $\displaystyle \mathrm{r}=0 \mathrm{~mm}$.

Mathematics · Differential Equations