- #1

Eveflutter

- 14

- 0

**1&2**The question:

The rate of deterioration of a product in a container is proportional to the amount of product present. At time t, the amount of product is x.

(i) State the diffrential equation relating x and t and solve the general solution to show that x=Ae

^{-kt}where A is an arbitrary contant and k is the contant of proportionality.

(ii) Before t=0, no product was present in the container. At t=0, P amount of the product was added to the container. When t=1, the amount of product in the container was Pq, where q is a constant such that 0<q<1. Show that x=Pq

^{t}.

(iii) When t=1 and again when t=2, another P amount of x was added to the container. Show that the amount of product in the container immediately after t=2 was P(q

^{2}+q+1)

**3**I got through with the first two parts:

(i) dx/dt=-kx

-> x=e

^{-kt+c}

-> x=Ae

^{-kt}, A=e

^{c}

(ii)when t=0, x=P

-> A=P

-> x= Pe

^{-kt}

when t=1, x=Pq

-> Pq=Pe

^{-kt}

-> q=e

^{-kt}

-> x=Pq

^{t}

I'm stuck at part (iii)

I tried reasoning it out but I don't know. Would the equation x=Pq

^{t}change since the intials are now changing twice (at t=1

*and*t=2)? But if so, how would I get the new ones from tha information?

Alright so as t approahes 1, x tends to Pq. At t=1 the amount of product changes to Pq+P. Then as t approaches 2, x deteriorates from Pq+P. But by how much?? Let's say it deteriorates till an amount B? Now at t=2, the amount of product goes to B+P.

I guess the P part of the required result (P(q

^{2}+q+1)) is that P in B+P. How do i get B?

I would really appreciate any help on this question, thank you!