Tin decay using separable equ's

[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!

 

[kuruman]

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"?

 

[kerimek]

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.