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: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 [tex]\frac{dV(t)}{dT} = A*V(t)^{2} \\[/tex] and V(0) = 900[tex]cm^{3}[/tex]where A is a real constant, it appears because it says that the change i volume (dV/dt) is "proportional" to [tex]V(t)^{2}[/tex]. 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.[tex]\frac{dr(t)}{dt}  = B*\sqrt{r(t)}[/tex]Here again appears a constant B for the "proportional" part. And i wrote the velocity as [tex]\frac{dr(t)}{dt}[/tex] "the rate of change of the position with respect to te time".