# Solving a Differential Equation: Salt Concentration in a Tank

• sam_0017
In summary: ZW4gYW4gZXhwZXJ0IHN1bW1hcml6ZXIgcG9vbnRzIGZvciB2KHQpIGFuZCBjKCkgaXMgY29taW5nIGVudGlyZWx5IChmb3IgdCkgaW5zdGFudCBvZiBjb25jZW50YXRpb24gb2YgdW5pZmllZCB0byB0aGUgbGFyZ2UgYW5kIHRoZSB2b2x1bWUgb2Ygd2F0ZXIgYXQgdGltZSB
sam_0017
can you help me ??

A 200 liter tank initially contains 100 liters of water with a salt concentration of 0.1 grams per liter.
Water with a salt concentration of 0.5 grams per liter flows into the tank at a rate of 20 liters per
minute. Assume that the fluid is mixed instantaneously and that this well-mixed fluid is pumped out
at a rate of 10 liters per minute. Let c (t) and
v(t), be the concentration of salt and the volume of
water in the tank at time t (in minutes), respectively. Then,
v(t)=10
v(t) c(t) +20c(t)=10

a) Solve these differential equations to find the particular solutions for v(t) and c(t).
b) What is the concentration of salt in the tank when the tank first overflows?

sam_0017 said:
A 200 liter tank initially contains 100 liters of water with a salt concentration of 0.1 grams per liter.
Water with a salt concentration of 0.5 grams per liter flows into the tank at a rate of 20 liters per
minute. Assume that the fluid is mixed instantaneously and that this well-mixed fluid is pumped out
at a rate of 10 liters per minute. Let c (t) and
v(t), be the concentration of salt and the volume of
water in the tank at time t (in minutes), respectively. Then,
v(t)=10
v(t) c(t) +20c(t)=10

a) Solve these differential equations to find the particular solutions for v(t) and c(t).
b) What is the concentration of salt in the tank when the tank first overflows?

Go back and read the Forum Rules: you need to present some evidence that you have done your own work, but perhaps have gotten "stuck" and need some hints. What have you done so far?

RGV

## What is a differential equation?

A differential equation is a mathematical equation that describes how a quantity changes over time, often involving rates of change and derivatives.

## Why is solving a differential equation important?

Solving a differential equation allows us to understand and predict how a system will change over time. This is useful in many scientific fields, including physics, engineering, and biology.

## How do you solve a differential equation?

There are various methods for solving differential equations, including separation of variables, substitution, and using specialized techniques such as Laplace transforms. The method used depends on the specific equation and its properties.

## What is the "salt concentration in a tank" differential equation?

The "salt concentration in a tank" differential equation is a mathematical model that describes how the concentration of salt in a tank changes over time. It takes into account factors such as the inflow and outflow of salt, as well as any reactions or mixing that may occur.

## What is the significance of solving the "salt concentration in a tank" differential equation?

Solving this differential equation allows us to determine the concentration of salt in the tank at any given time, as well as how it changes over time. This can be useful in industries such as water treatment and chemical engineering, where precise control of salt concentration is important.

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