Question

Can someone explain why, in exponential decay, P=Po(1/2)^(t/h), where P = remaining amount of something after t years, Po = original amount, t = number of years, and h = half life in years, why the exponent is (t/h)?

Answer 1

sure

Answer 2

first of all you need to know what "half life" means. is this clear or no?

Answer 3

yes half life is clear

Answer 4

ok then lets say for convenience that the half life is 5 days, and you start with say 16 grams

Answer 5

we start counting at time t = 0 and then after 5 days we have 8 grams, after 10 days we have 4 grams, after 15 days we have 4 grams etc

Answer 6

little chart looks like this

t 0 5 10 15 20
g 16 8 4 2 1

Answer 7

the second row of the chart is a geometric sequence where the first term is 16 it looks like this'
\[16, 16\times (\frac{1}{2}), 16\times (\frac{1}{2})^2, 16\times (\frac{1}{2})3,16\times (\frac{1}{2})^4...\]

Answer 8

typo there but i won't fix it should be cubed

Answer 9

now then
\[t=5\] you get
\[16\times (\frac{1}{2})^1\] and when
\[t=10\] you get
\[16\times (\frac{1}{2})^2\] and wehn
\[t=15\] you get
\[16\times (\frac{1}{2})^3\]

Answer 10

so now the question is "what do you get if t is not a multiple of 5?"

Answer 11

and the answer is, just divide by 5 right? if t = 1 you DIVIDE BY 5 and get an exponent of 1 if t is 15 you DIVIDE BY 5 and get an exponent of 3, and if t is 20 you DIVIDE BY 5 and get an exponent of 4

Answer 12

so in general if you have time and it does not happen to be a multiple of 5, you can still divide by 5. for example if i want to know how much i have in say 22 days i would compute
\[16\times (\frac{1}{2})^{\frac{22}{5}}\]

Answer 13

that is my best explanation. if it is not clear, post again and probably some one can explain better

Answer 14

this was great thanks!

Answer 15

yw