Question

A spherical raindrop evaporates at a rate proportional to its surface area. Write down a differ- ential equation for its volume as a function as a function of time. Solve this equation, and find the constant of proportionality if a 1cm3 raindrop takes 10 seconds to evaporate.

Answer 1

Answer:

$\dfrac{dV}{dt} =-4 \pi k (\dfrac{3}{4 \pi})^\dfrac{2}{3} }}V^{\dfrac{2}{3}}$

k = 3.022

Step-by-step explanation:

Given that:

A spherical raindrop evaporates at a rate proportional to its surface area.

The surface area SA of a spherical object is given by the relation:

SA = 4πr²

Write down a differ- ential equation for its volume as a function as a function of time.

So; to differentiate Volume (V) in respect to time (t) ;then:

$\dfrac{dV}{dt} =-k( 4 \pi r^2)$

Likewise; we know known that the volume of a sphere V = $\dfrac{4}{3} \pi r^3$

Thus, from above;

$3V = 4 \pi r^3$

$\dfrac{3V}{4 \pi} = r^3$

$r^3 = (\dfrac{3}{4 \pi })^{\dfrac{2}{3}}V^{\dfrac{2}{3}}$

$r^2 = 4 \pi( \dfrac{3}{4 \pi})\dfrac{2}{3}V\dfrac{2}{3}$

Thus; solving the differential:

$\dfrac{dV}{dt} =-k( 4 \pi * 4 \pi( \dfrac{3}{4 \pi})\dfrac{2}{3}V\dfrac{2}{3})$

$\dfrac{dV}{dt} =-4 \pi k (\dfrac{3}{4 \pi})^\dfrac{2}{3} }}V^{\dfrac{2}{3}}$

So;

we are to find the constant proportionality K

If Volume V = 1 cm³ and the time = 10 sec

$\dfrac{1}{10} =-4 \pi k (\dfrac{3}{4 \pi})^\dfrac{2}{3} }}(1)^{\dfrac{2}{3}}$

0.1 =  - 4π k (0.3848 × 1)

0.1 =  - 4π k × 0.3848

4π k = 0.3848/0.1

4π k  = 3.848

k = 3.848/4π

k = 3.022