Differential equation question?

Question

Please someone help.... I'm very lost!!

write the differential equation that fits the physical description.

1. A (spherical) mothball looses volume by evaporation at a rate proportional to its instantaneous area. Hint: you must express the surface area of a sphere in terms of its volume.

2. The velocity at time t of a particle moving along a straight line is proportional to the fourth power of its position x.

I appreciate any help, hints or ideas on how to approach these problems!!

Answer 1

The volume V of the sphere of radius r is given by

V = (4/3)πr³

The surface area A is given by

A = 4πr²

The hint says to express A in terms of V. Solve V for r, then put that into the expression for A.

V = (4/3)πr³

[3V/(4π)]^(1/3) = r

A = 4πr² = 4π [3V/(4π)]^(2/3) = (36π)^(1/3) V^(2/3)

So now,

dV/dt = kA = k(36π)^(1/3) V^(2/3)

where k is the proportionality constant.

2)

velocity is the derivative of position x with respect to time. So

dx/dt = kx^4

where again k is the proportionality constant.