# Modeling Using Differential Equations

• Apr 16th 2010, 12:12 AM
iyppxstahh
Modeling Using Differential Equations
I'm having some trouble with this assignment:

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.

1. Write a differential equation that models this situation, Let V represent the volume (in gallongs) in the tank and t represent the time (in minutes).
- dV/dt=k*sqrtV

2. Solve for the general solution (do not solve for V).
-I WAS going to solve for V, but it says not to. So I thought maybe it's meaning to find the integral? Which is :
(2*k*v^(3/2)/3) = t

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

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

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

I think I might be able to do this if I knew number 2. What does it mean by "do not solve for V"? Does it mean don't evaluate the equation?
If it's okay, could you help me on 2 first? I want to know the correct answer before I solve the other ones based on it.
Thanks.
• Apr 16th 2010, 03:32 AM
HallsofIvy
Quote:

Originally Posted by iyppxstahh
I'm having some trouble with this assignment:

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.

1. Write a differential equation that models this situation, Let V represent the volume (in gallongs) in the tank and t represent the time (in minutes).
- dV/dt=k*sqrtV

2. Solve for the general solution (do not solve for V).
-I WAS going to solve for V, but it says not to. So I thought maybe it's meaning to find the integral? Which is :
(2*k*v^(3/2)/3) = t

How did k go from multiplying on the right to multiplying on the left? And you forgot the constant of integration.

Quote:

3. Use the inital condition to find the constant of integration, then write the particular solution (do not solve for V).
$\displaystyle \frac{2}{3}V^{3/2}= kt+ C$. When t= 0 V= 25. What is C?

[quote]4. Use the second condition to find the constant of proportion.
When t= 3, V= 20.25. What is k?

Quote:

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

I think I might be able to do this if I knew number 2. What does it mean by "do not solve for V"? Does it mean don't evaluate the equation?
By integrating, you got an equation involving V. This just means "don't solve that equation for V.

Quote:

If it's okay, could you help me on 2 first? I want to know the correct answer before I solve the other ones based on it.
Thanks.
• Apr 16th 2010, 04:17 AM
mahm32
use the separation method of the differential equations
-dV/sqrt v =k dt then integrate both sides . this will get you a general equation then use your given information to find K and C
• Apr 19th 2010, 07:35 AM
iyppxstahh
Ohhhhkay, I am able to solve these now. Thanks!