Pre-Cal word problem.

A sodium isotope, Na24, has a half-life of 15 hours. Determine the amount of sodium that remains from a 4g samples after:

i) 45 hours

ii) 100 hours

iii) 5 days

Question

Pre-Cal word problem. 

A sodium isotope, Na24, has a half-life of 15 hours. Determine the amount of sodium that remains from a 4g samples after: 

i) 45 hours

ii) 100 hours

iii) 5 days

anonymous · Answer

100

anonymous · Answer

how did you get that and which question is that your answer to?

polaris_s0i · Answer

First you need to come up with the formula for the half life.

Since a half life basically says the amount of mass that will remain is cut in half ever 15hrs, you want to end up with this:

(0h, 4g)
(15h, 2g)
(30h, 1g)
(45h, 0.5g)
...

that should be enough to give you the formula.

anonymous · Answer

so there isn't already a formula for half lives?

polaris_s0i · Answer

sure there is

anonymous · Answer

I'm so sorry I know that it is decreasing but it decreases from 2 to 1 to .5.. so how would I formulate that to find what it is for 45, 100 hours and 5 days?

polaris_s0i · Answer

so depending on you're "scale" for time (I'm using the hours as given)

$$m = \frac{4}{2^{\frac{t}{15}}}$$

anonymous · Answer

is the t/15 an exponent?

polaris_s0i · Answer

yes

anonymous · Answer

so would the answer to 1 for 45 hours be 1/2?

polaris_s0i · Answer

yes.
now you should just have to plug in the other times.

anonymous · Answer

hes wrong 5 days
im in AP CaL

anonymous · Answer

thanks so much!! one last question, on #2 100 hours, the exponent is 100/15 which in decimal form is 6.666667.. 

would i use that for the exponent or use the fraction 20/3?

anonymous · Answer

@naahhbass what do you mean 5 days??

polaris_s0i · Answer

just so you're aware of why this equation makes sense:

0 -> m
1 -> 1/2 *  m
2 -> 1/2(1/2 * m) = 1/4 *m
3 -> 1/2(1/2(1/2 * m) = 1/8 * m

t -> 1/2^t * m

polaris_s0i · Answer

I don't know where he's getting that from, but this is from wikipedia:

$$N(t) = N_0 \left(\frac {1}{2}\right)^{t/t_{1/2}}$$

where
N0 is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc.),
N(t) is the quantity that still remains and has not yet decayed after a time t,
t1/2 is the half-life of the decaying quantity,
τ is a positive number called the mean lifetime of the decaying quantity,
λ is a positive number called the decay constant of the decaying quantity.

anonymous · Answer

thank you very much! :)