Answer:
[tex]\dfrac{dV}{dt} =-4 \pi k (\dfrac{3}{4 \pi})^\dfrac{2}{3} }}V^{\dfrac{2}{3}}[/tex]
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:
[tex]\dfrac{dV}{dt} =-k( 4 \pi r^2)[/tex]
Likewise; we know known that the volume of a sphere V = [tex]\dfrac{4}{3} \pi r^3[/tex]
Thus, from above;
[tex]3V = 4 \pi r^3[/tex]
[tex]\dfrac{3V}{4 \pi} = r^3[/tex]
[tex]r^3 = (\dfrac{3}{4 \pi })^{\dfrac{2}{3}}V^{\dfrac{2}{3}}[/tex]
[tex]r^2 = 4 \pi( \dfrac{3}{4 \pi})\dfrac{2}{3}V\dfrac{2}{3}[/tex]
Thus; solving the differential:
[tex]\dfrac{dV}{dt} =-k( 4 \pi * 4 \pi( \dfrac{3}{4 \pi})\dfrac{2}{3}V\dfrac{2}{3})[/tex]
[tex]\dfrac{dV}{dt} =-4 \pi k (\dfrac{3}{4 \pi})^\dfrac{2}{3} }}V^{\dfrac{2}{3}}[/tex]
So;
we are to find the constant proportionality K
If Volume V = 1 cm³ and the time = 10 sec
[tex]\dfrac{1}{10} =-4 \pi k (\dfrac{3}{4 \pi})^\dfrac{2}{3} }}(1)^{\dfrac{2}{3}}[/tex]
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
