Let's break down this problem into the two parts given.
### Part (a): Calculating [tex]\( L \)[/tex]
The equation given is [tex]\( L = \frac{a}{1 - p} \)[/tex].
- [tex]\( a \)[/tex] is the dosage, which is 500 ml.
- [tex]\( p \)[/tex] is the percent that remains in the bloodstream, which is 0.4 (or 40%).
Substitute the values into the formula:
[tex]\[
L = \frac{500}{1 - 0.4}
\][/tex]
Now, calculate the denominator:
[tex]\[
1 - 0.4 = 0.6
\][/tex]
So the formula becomes:
[tex]\[
L = \frac{500}{0.6}
\][/tex]
By performing the division:
[tex]\[
L \approx 833.33 \, \text{ml}
\][/tex]
Thus, the amount that the medication levels off to is approximately [tex]\( 833.33 \)[/tex] ml.
### Part (b): Calculating the amount of medication in the bloodstream after 8 doses
The equation given is [tex]\( A(n) = (a - L) \cdot p^{n-1} + L \)[/tex].
- [tex]\( a \)[/tex] is the dosage, which is 500 ml.
- [tex]\( L \)[/tex] is the amount that the medication levels off to, which we found to be approximately [tex]\( 833.33 \)[/tex] ml.
- [tex]\( p \)[/tex] is the percent that remains in the bloodstream, which is 0.4.
- [tex]\( n \)[/tex] is the number of doses, which is 8 in this case.
First, calculate [tex]\( a - L \)[/tex]:
[tex]\[
a - L = 500 - 833.33 \approx -333.33
\][/tex]
Then calculate [tex]\( p^{n-1} \)[/tex]:
[tex]\[
p^{7} = 0.4^{7}
\][/tex]
Finding [tex]\( 0.4^7 \)[/tex], we get approximately:
[tex]\[
0.4^7 \approx 0.0016384
\][/tex]
Substitute these values into the formula:
[tex]\[
A(8) = (-333.33 \cdot 0.0016384) + 833.33
\][/tex]
Calculate the first part of the expression:
[tex]\[
-333.33 \cdot 0.0016384 \approx -0.5461333332
\][/tex]
Add this to [tex]\( L \)[/tex]:
[tex]\[
A(8) = -0.5461333332 + 833.33 \approx 832.7872
\][/tex]
Thus, the amount of medication in the bloodstream after 8 doses is approximately [tex]\( 832.79 \)[/tex] ml.
