Question

Write the differential equation model that fits the given description. (a) The rate of change of the volume of a snowball (due to melting) is proportional to the square of the volume at time t. Initially, the snowball has a volume of 900 cm3 . (b) For an insect moving along some path, the velocity at time t is proportional to the square root of its position.

Answer 1

Answer:

a) he rate of change of the volume of a snowball (due to melting) is proportional to the square of the volume at time t. Initially, the snowball has a volume of 900 cm3

$\frac{dV(t)}{dT} = A*V(t)^{2}$ and V(0) = 900$cm^{3}$

where A is a real constant, it appears because it says that the change i volume (dV/dt) is "proportional" to $V(t)^{2}$. Furthermore, we should assume that A is a negative number, because the volume of the snowball will decrease as the time pasese by.

(b) For an insect moving along some path, the velocity at time t is proportional to the square root of its position.

$\frac{dr(t)}{dt} = B*\sqrt{r(t)}$

Here again appears a constant B for the "proportional" part. And i wrote the velocity as $\frac{dr(t)}{dt}$ "the rate of change of the position with respect to te time".