Modeling Using Differential Equations

olicoh
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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?
 
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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
 
miglo said:
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
My answer: 2√(V)=kt+10


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
My answer: k = -2


5) Find the volume at t = 5 minutes. Round your answer to two decimal places.


My attempt at the solution: 2√(V)=(-2)(5) + 10 --> 2√(V)=0
My answer: V(5)=0

^^^
For number 5... I'm not sure my answer is right. Using common sense, I don't think it's possible for the tank to be at zero gallons at 5 minutes. I think I might have done something wrong at number 4.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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