Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
If right circular cone with radius 18 & height 27 contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant = $k > 0$). If volume of liquid is $V$ & $r$ is radius of surface of liquid left, then:
$\frac{1}{r^2}\frac{dV}{dr} = \frac{3\pi}{2}$
$\frac{1}{r^2}\frac{dr}{dt} = -k\pi$
Radius as function of time $r(t) = \frac{-2k}{3}t + c$
Total time taken to empty the cone is $\frac{3}{2}$ unit
Step-by-Step Solution
Key Concept: Related rates problems require expressing volume in terms of a single variable using given constraints, then differentiating with respect to time.
Given $V = \frac{1}{3}\pi r^2 h$ and $\frac{h}{r} = \frac{27}{18}$, we substitute to get $V = \frac{1}{3}\pi r^2 \cdot \frac{27r}{18} = \frac{27\pi r^3}{54} = \frac{\pi r^3}{2}$. Differentiating with respect to time: $\frac{dV}{dr} = \frac{3\pi r^2}{2}$ and using chain rule $\frac{dV}{dt} = \frac{dV}{dr}\cdot\frac{dr}{dt}$, we get $\frac{dV}{dt} = \frac{3\pi r^2}{2}\cdot\frac{dr}{dt} = \frac{3\pi}{2}$.
Correct Answer: 1,3