# Tin decay using separable equ's

• jenettezone
In summary, the amount of decayed tin p(t) can be found by using the differential equation dp/dt = kq(t)(M-q(t)), where q(t) represents the amount of tin left and M is the initial amount of tin. This equation shows that the rate of decay is proportional to the product of the amount of tin left and the amount already decayed.
jenettezone

## Homework Statement

A tin organ pipe decays with age as a result of a chemical reaction that is catalyzed by the decayed tin. as a result, the rate at which the tin decays is proportional to the product of the amount of tin left and the amount that has already decayed. Let M be the total amount of tin left before any has decayed. Find the amount of decayed tin p(t).

## Homework Equations

"Find the amount of decayed tin p(t)." → p(t)=decayed tin,

"Let M be the total amount of tin left before any has decayed." → amount of tin left=M-p(t)?

## The Attempt at a Solution

"the rate at which the tin decays is proportional to the product of the amount of tin left and the amount that has already decayed." → dp/dtαM-p(t)αp(t)

I am not sure if my interpretation of the data is correct. or if it is, i do not know what to do with it to get an equation for p(t). please help!

Let q(t) = amount of tin left. Then
M-q(t) = amount of tin decayed and
q(t)*(M-q(t))=product of amount left and amount already decayed.

Can you write a differential equation saying "the rate at which the tin decays is proportional to the product of the amount of tin left and the amount that has already decayed"?

Your interpretation is correct. To get an equation for p(t), we can use the separable equation method. We can rewrite the equation as dp/dt = kp(t)(M-p(t)), where k is a constant of proportionality. We can then separate the variables and integrate both sides to get p(t) as a function of t:

∫dp/p(t) = ∫k(M-p(t))dt

ln(p(t)) = kt + C

p(t) = e^(kt+C) = Ae^kt

Where A is a constant of integration. We can use the initial condition that p(0) = 0 (since no tin has decayed at t=0) to solve for A:

p(0) = Ae^k(0) = 0

A = 0

Therefore, our final equation for p(t) is:

p(t) = 0

This means that all the tin will decay eventually and there will be no tin left.

## 1. What is the process of tin decay using separable equations?

Tin decay using separable equations is a method used in scientific research to study the decay of tin atoms over time. It involves the use of mathematical equations that can be separated into two parts, making it easier to solve and analyze the decay process.

## 2. Why is tin decay using separable equations important?

Tin decay using separable equations is important because it allows scientists to understand and predict the rate of decay of tin atoms, which is useful in various fields such as archaeology, geology, and nuclear physics. It also helps in developing new technologies and improving existing ones.

## 3. What are the main components of separable equations used in tin decay?

The main components of separable equations used in tin decay are the rate of decay (represented by the letter k), the initial amount of tin atoms (represented by the letter A), and time (represented by the letter t). These components are used in the equation A = A0e^-kt, where A0 is the initial amount of tin atoms and e is the base of natural logarithm.

## 4. How can separable equations be applied in studying tin decay?

Separable equations can be applied in studying tin decay by using them to model the decay process and calculate the amount of tin atoms remaining at different time intervals. This information can then be compared to actual observations to validate the accuracy of the model and gain a better understanding of the decay process.

## 5. Are there any limitations to using separable equations in studying tin decay?

Yes, there are limitations to using separable equations in studying tin decay. These equations assume that the rate of decay is constant over time, which may not always be the case. They also do not take into account external factors that may affect the decay process, such as temperature and pressure. Therefore, it is important to use these equations in conjunction with other methods and techniques to get a more accurate understanding of tin decay.

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