calculus (don't get the question)?

Question

*Write down a differential equation to describe the following situations :

a) A pie just been taken out of the oven. The rate of decrease of the temperature T of the pie is proportional to the difference between T and the temperature T of the surrounding air.

b)John's water supply is stored in an upright cylindrical tank. Unfotunately the tank has developed a small hole in its base and water is leaking out at a rate proportional to the square root of the volume of water remaining.

Answer 1

a. the rate of change of the temperature of the pie is dT/dt where t is the time.

Proportional to the difference in temperature between the pie and that of the surrounding air (call Q the temperature of the surrounding air). This is just: T - Q

so dT/dt = -(T - Q) (the temperature of the pie is falling)

b. dV/dt is the rate the water is leaking if V is the volume of water in the tank (the time rate of change of the volume of water in the tank)

This leak rate is proportional to SQRT(V)

dV/dt = SQRT(V)

Answer 2

a) ok, so the variables are T=pie temp, Ts=air temp, t=time;

for these questions, you really need to just interpret the wording correctly.

''rate of temp decreases'' -- it's differential of temperature with respect to time! or dT/dt

''proportional to difference between pie temp and air temp'' -- to use k as the proportion constant; k(T - Ts)

So putting this together;

dT = k(T - Ts)
dt

b) variables; V= volume, t = time
The process is much the same. This is as close to english that math gets :) sortof.

dV = k(V^1/2)
dt

hope that helps. study hard to remember stuff your teacher/lecturer says! differentials can get messssssy