Please help with half life question

[michc]

I need help with this half life question as my final answer does not make sense for what I think the answer should be. The question is "The drug Prozac has a half life of about 3 days. a)what percentage of a dose remains in the body after 1 day?
b)after 1 week?
For a) 1/2=1e^k3
ln1/2=lne^k3
ln1/2=k3
(ln1/2)diveded by3=k
then k =-0.23??

for b I did the same process of equations for 1/2=7e^k3 and got -0.12??

If you could help point out my error it would be greatly appreciated!!

 

[tkhunny]

1/2 of 1 is 1/2

1/2 of 7 is 3½

Why would you expect the same answer when you are working a different problem.

-0.231049 is good.

You will get the same answer when you try 3½ = 7*e^(-3k)

Work with one half of what you started with, not just 1/2.

 

[michc]

I'm sorry I should clarify more that I am not provided with an initial amount and I feel that I made a mistake becuase I think after one day more than 23% would remain in the body

 

[mmm4444bot]

The expression -k/3 does not represent the remaining dose-percentage after 1 day.

For this exercise, I would first determine a function f(t) that gives the remaining percentage in terms of elapsed days t.

Then, it's simply a matter of evaluating f(1) and f(7).

f(t) = A * e^(k * t/3)

where A is the starting dose, f(t) is the amount remaining, and t is the number of elapsed days.

You already found the value of k/3.

You could let A equal 100 units of Prozac. (That's enough to make everybody happy.)

 

[lookagain]

For a) 1/2 = 1e^(k*3)

ln(1/2) = lne^(3k)

ln(1/2) = 3k

ln(1/2) divided by 3 = k

then k = -0.23??

michc,

be sure to use grouping symbols as shown in the above places, if not in additional places.
Also, for increased readability, space out your work horizontally and vertically as shown
in the above.

I find it convenient to memorize and use these:

\(\displaystyle {\text{half-life} \ = \ \frac{ln(.5)}{k} \ \ and\)


\(\displaystyle k \ = \ \frac{ln(.5)}{\text{half-life}}\)

 

[tkhunny]

I'm sorry I should clarify more that I am not provided with an initial amount and I feel that I made a mistake becuase I think after one day more than 23% would remain in the body

This clarification is not needed. When you started, you used "1" and moved to "1/2". That is perfect.

When you tried again, you started as "7" and moved to "1/2". That's no good.

 

[Denis]

Perhaps making up a simple case will illustrate.
Say we have a crooked bank that pays -12% annually: when will a $1 deposit reduce to 50 cents ?
Formula: (1 + i)^n = .5 ; since i = -12, then:
(1 + (-.12))^n = .5 : .88^n = .5 : n = log(.5) / log(.88) = 5.42227...so in not quite 5 1/2 years

Here's how "it looks":

YEAR     INTEREST    BALANCE
0                    1.00000
1        -.12000      .88000
2        -.10560      .77440
3        -.09293      .68147
4        -.08177      .59970
5        -.07196      .52774
5.42227  -.02774      .50000

Your problem works the same way...

 

[tkhunny]

CanI get a loan like that?!

 

[Denis]

Due to your credit rating, you'll need to put 2 bucks as security for a one buck loan!!