Question

What is the half life of a radioative substance that decays by 6% in 7 years?

Answer 1

half life is the period of time it takes to decay 50% of its original value. so what ration of 50% is 6% and apply that ratio to 7

Answer 2

ummmm 50/6?

Answer 3

So I make it about 58.3333

Answer 4

yep 50/6 * 7

Answer 5

Ohh! Thanks!

Answer 6

hmm that is not what i got. let me try again.

Answer 7

i got 78.4

Answer 8

i think this problem is a little more complicated that ratios. the general formula for decay is
\[A=A_0e^{rt}\] where r is the rate of decay (r will be negative) t is time, \[A_0\] is the initial amount and \[A\] is what you have after t years. we do not know r, but we know that when t = 7 you have decreased by 6%. decreasing by 6% means you have 94% =.94 of the original amount so set
\[.94=e^{7r}\] and solve for r
\[7r = ln(.94)\]
\[r=\frac{ln.94}{7}=-.008839\] rounded
now half life means you have half the original amount so set
\[\frac{1}{2}=e^{-.00839t}\] or better still (to avoid rounding error)
\[\frac{1}{2}=e^{\frac{ln(.94)}{7}t}\] and solve for t
\[ln(\frac{1}{2})=\frac{ln(.94)t}{7}\]
\[t=\frac{7ln(\frac{1}{2})}{ln(.94)}=78.4\] rounded

Answer 9

i could be wrong of course, but i am fairly sure this is correct.

Answer 10

ah yes what was I thinking. but isn't decay \[e^{-rt}\]
and not \[e^{rt}\] because that's growth not decay
so taking the log of both sides should give \[-7r=\ln(0.94)\]

the rest follows?

Answer 11

I agree with the answer given. here is a doc to clarify the above comment. sorry for the misunderstanding

Answer 12

you can write \[A=A_0e^{-rt}\] it is just a matter of preference. my 'r' was negative.
if you write it your way r is positive. it makes no difference.

Answer 13

there is another way to solve this problem that amounts to the same thing. since you know it decreases by 6% in 7 years you could write the formula is
\[A=A_0(.94)^{\frac{t}{7}}\]
then set
\[\frac{1}{2}=(.94)^{\frac{t}{7}}\]
and solve for t, via
\[ln(\frac{1}{2})=ln((.94)^{\frac{t}{7}})=\frac{t}{7}ln(.94)\]
and still get
\[t=\frac{7ln(\frac{1}{2})}{ln(.94)}\]

Answer 14

Nice :D