# Differential Equations mixing sugar and water

• hocuspocus102
In summary, the problem involves a tank containing 2860 L of water and a solution with a concentration of 0.04 kg/L of sugar entering and exiting at a rate of 5 L/min. Through integration, the amount of sugar in the tank after t minutes is found to be 114.4 - 114.4e^(-t/572). However, despite repeated attempts, the answer is not accepted by the online system.
hocuspocus102

## Homework Statement

A tank contains 2860 L of pure water. A solution that contains 0.04 kg of sugar per liter enters a tank at the rate 5 L/min The solution is mixed and drains from the tank at the same rate. Find the amount of sugar in the tank after t minutes.

## The Attempt at a Solution

Obviously it starts with 0 kg sugar and an initial condition at time t = 0.
The rate then would be equal to (the rate entering) - (the concentration exiting).
Entering = .04*5 = .2
Exiting = (Y*5)/2860 = Y/572

dY/dt = .2 - Y/527
dY/(.2 - Y/572) = dt
integrate both sides
-572ln(.2-Y/572) = t + C
simplify
ln(.2 - Y/572) = -t/572 + C
.2 - Y/572 = e^(-t/572+C)
-Y/572 = Ce^(-t/572) - .2
Y = Ce^(-t/572) + 114.4
initial value t = 0 and Y = 0
0 = Ce^(-0/572) + 114.4
C = -114.4
so Y = 114.4 - 114.4e^(-t/572)

Is this right because I've done it a bunch of times and keep getting the same answer, but it's an online problem and it tells me that it's wrong and I can't figure out what I did incorrectly. Thanks!

I don't see anything wrong with your solution. Beats me why the online thing won't take it.

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It is used to model the relationship between a quantity and how it changes over time or space.

## 2. How are differential equations used to mix sugar and water?

Differential equations can be used to describe the process of mixing sugar and water by modeling the rate of change of the concentration of sugar in the mixture over time. This allows us to predict how long it will take for the sugar to fully dissolve in the water.

## 3. What are the variables involved in the differential equation for mixing sugar and water?

The variables in the differential equation for mixing sugar and water are time and concentration of sugar. Time represents the independent variable, while the concentration of sugar represents the dependent variable that changes over time.

## 4. How is the initial concentration of sugar in the water determined in the differential equation?

The initial concentration of sugar in the water is determined by the initial conditions of the system. These can be set by the experimenter or estimated based on previous experiments or knowledge of the system.

## 5. Can differential equations be solved analytically for mixing sugar and water?

Yes, depending on the specific differential equation and initial conditions, it may be possible to solve the equation analytically. However, in most cases, numerical methods are used to approximate the solution due to the complexity of the equation.

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