exponential growth and decay (1 Viewer)

[jkingx]

12) the rate of decay of a radioactive substance is proportional to the amount of the substance available at any time t.
1) if half of the original amount is decomposed after 2000 years, find the yearly decay rate.
2) if the original amount is 2.5g, find the amount present after 500 years.
3) what is the rate of decay after 1200 years?

just learn this topic 2day n stuck on this HW question

 

[tessayu]

12) the rate of decay of a radioactive substance is proportional to the amount of the substance available at any time t.

your equations are : M=M0e^(-kt) and dM/dT= -kM

1) if half of the original amount is decomposed after 2000 years, find the yearly decay rate.

you have to find the initial mass first, which is M0 and because they have not given you a specific number, you work with percentages ( original amount =100 ), so:

let M=100, t=0
ie M0e^(-k x 0) = 100
and since anything to the power of 0 equals 1 when applied to any numeral, e^(-k x 0)=1
thus, M0= 100

now we have to find k, which is the constant and we just substitute everything we know and apply it to to formula

let M0=100, t=2000 and M=50
ie 100e^(-2000k)= 50

e^(-2000k)= 1/2

-2000k= -ln2 ( ln1/2 = -ln2)

thus, k= ln2/2000

after 2000 years,
M= 100e^(-ln2/2000 x 2000)
thus M=50

to find the rate at which it is decaying, we substitute our numbers into the derivative formula
dM/dt= -ln2/2000 x 50
= 1.7 x 10^-2 g/year

2) if the original amount is 2.5g, find the amount present after 500 years.

let M=2.5 , t=0

ie M0e^0= 2.5
thus M0=2.5


let M0= 2.5, t= 500

ie M= 2.5e^(-ln2/2000 x 500)
= 2.1 g (1dp)

3) what is the rate of decay after 1200 years?

let t= 1200, M0=2.5
ie M= 2.5e^(-ln2/2000) x 1200)
= 1.65g (2dp)

dM/dt= -ln2/2000 x 1.65
= 5.7 x 10^-4 g/year

just learn this topic 2day n stuck on this HW question

I hope its right!~ Double check of course if i havn't explained clearly, feel free to ask

sorry about the format: ^ means "to the power of"

 

[MaNiElla]