# Modeling Using Differential Equations

## Homework Statement

Suppose the rate at which the volume in a tank decreases is proportional to the square root of the volume present. The tank initially contains 25 gallons, but has 20.25 gallons after 3 minutes. Solve for the general solution (do not solve for V).

## The Attempt at a Solution

dV/dt = k√(V)

That's as far as I got. I know I have to "separate" the variables and whatnot, but there is no 't' to separate and differentiate from. I guess the equation would be: ∫1/√(V) dV = k∫ ___ dt
So my question is, since there is no 't' in the equation, what do I differentiate instead?

your setup is correct, k is just some constant and whenever you integrate a constant with respect to x, t, z, etc. your left with k multiplied by the variable you integrated with respect to
so in your case, for the RHS youll get kt+C

your setup is correct, k is just some constant and whenever you integrate a constant with respect to x, t, z, etc. your left with k multiplied by the variable you integrated with respect to
so in your case, for the RHS youll get kt+C
That's what I thought. I just wanted to double check. Thanks!

I have another question (actually 3 questions):

3) Use the initial condition to find the constant of integration, then write the particular solution (do not solve for V).

Attempt at solution: 2√(25)=k(0) + C

4) Use the second condition to find the constant of proportion.

Attempt at solution: 2√(20.25)=k(3) + 10 --> 4=3k+10 --> -6=3k