Let's evaluate the given function to determine the initial amount of the drug injected into the patient and the amount of the drug remaining in the bloodstream after 8 hours.
The function modeling the drug amount is given by:
[tex]\[ D(h) = 5 e^{-0.45 h} \][/tex]
First, we need to find the initial amount injected, which is the value of [tex]\( D(h) \)[/tex] when [tex]\( h = 0 \)[/tex].
### Finding the Initial Amount
1. Substitute [tex]\( h = 0 \)[/tex] into the function:
[tex]\[ D(0) = 5 e^{-0.45 \cdot 0} \][/tex]
2. Simplify the exponent:
[tex]\[ D(0) = 5 e^{0} \][/tex]
3. Recall that [tex]\( e^0 = 1 \)[/tex]:
[tex]\[ D(0) = 5 \cdot 1 \][/tex]
4. Simplify the multiplication:
[tex]\[ D(0) = 5 \][/tex]
So, the initial amount of the drug injected is [tex]\( 5.0 \)[/tex] milligrams.
### Finding the Amount After 8 Hours
Next, we need to calculate the amount of the drug in the bloodstream after 8 hours, which is the value of [tex]\( D(h) \)[/tex] when [tex]\( h = 8 \)[/tex].
1. Substitute [tex]\( h = 8 \)[/tex] into the function:
[tex]\[ D(8) = 5 e^{-0.45 \cdot 8} \][/tex]
2. Calculate the exponent:
[tex]\[ D(8) = 5 e^{-3.6} \][/tex]
3. Evaluate the exponential function [tex]\( e^{-3.6} \)[/tex]:
[tex]\[ e^{-3.6} \approx 0.02732 \][/tex]
4. Multiply by the constant 5:
[tex]\[ D(8) = 5 \cdot 0.02732 \][/tex]
5. Simplify the multiplication:
[tex]\[ D(8) \approx 0.1366 \][/tex]
Rounding to the nearest hundredth:
[tex]\[ D(8) \approx 0.14 \][/tex]
So, the amount of the drug in the bloodstream after 8 hours is approximately [tex]\( 0.14 \)[/tex] milligrams.
### Final Answer
- Initial amount: [tex]\( 5.0 \)[/tex] milligrams
- Amount after 8 hours: [tex]\( 0.14 \)[/tex] milligrams
