Question

The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time t and has a half life of 3.3 hrs. If 1 gram of this isotope is present initially, howlong will it take for 90% of the lead to decay?

Answer 1

i already know that the proportion is

\[\frac{dA}{dt}=kA\]

Answer 2

dividing by A and multiplying by dt you get

\[\frac{dA}{A}=kdt\]

Answer 3

if you integrate you get

\[Ln(A)=kt+c\]

Answer 4

\[A=ce^{kt}\]

Answer 5

using the initial condition A(0)=1

Answer 6

\[c=A_0\]

Answer 7

yes it is A(0)=c which is 1

\[A=e^{kt}\]

correct so far?

Answer 8

\[90\%=\frac{A}{A_0}=e^{-kt}\]

Answer 9

the second statement says that after 3.3 years it has a half life

is that saying

P(3.3)=.5(A(0)

Answer 10

that's is what i did

take the ln of both sides

ln(.5)=3.3k

Answer 11

ln(.5)/3.3=k

Answer 12

\[A=e^{\frac{ln.5)t}{3.3}}\]

Answer 13

k or -k,

\[\frac{\text dA}{\text dt}=-kA\] for decay

Answer 14

the book uses kA for hte decay example

Answer 15

so i assumed that it doens't matter due to it being a constant?

Answer 16

\[.1=e^{\frac{ln(.5)t}{3.3}}\]

Answer 17

take ln of both sides? then mutliply by the reciprocal?

Answer 18

i swear i did the sam ething and got the wrong answer the first time -.- but yes it's correct 10.9 or 11 hrs

Answer 19

i must have put it wrong in the calculator somehow

Answer 20

or i tried to simplify the right side way too much

Answer 21

i think thats what i did i thought the t was added so it was ln(.5)+t and split it

Answer 22

want to explain kirchoff's law. I was sick the day my teacher expained how it works

Answer 23

Not an electrical engineer =P lol

Answer 24

kirchoff's law

the sum of current flowing into a junction
=
the sum of current flowing out of the junction

Answer 25

yes i mean't the variables R, L, q/c

Answer 26

what are you asking ?

Answer 27

\[Ri+\frac{q}{C}=E(t)

Answer 28

\[Ri+\frac{q}{C}=E(t)\]

Answer 29

how these formulas are derived. This is probablylike the first time i've ever seen circuits before so i don't know how they relate to eachother

Answer 30

Driven Harmonic motion is
\[m\ddot x+c\dot x+kx=F(t)\]
which comes from the Linear Differential Equation
\[y''(x)+P(x)y=Q(x)\]
In the case of an electric circuit
\[L\frac{\text d^2 q}{\text dt^2}+\frac{\text d q}{\text dt}R+\frac qC=V(t)\]
if the inductance is zero\[IR+\frac{q}{C}=V(t)\]

Answer 31

**Second order linear differential equation

\[P(x)\frac{\text d^2 y}{\text dx^2}+Q(x)\frac{\text d y}{\text dx}+R(x)y=G(x)\]