The equation provided models the number of milligrams $D(h)$ of a drug in the bloodstream $h$ hours after injection:

$D(h) = 50 e^{-0.55h}$

1. Initial Amount Injected

The initial amount of the drug injected corresponds to $D(0)$, i.e., the amount at $h = 0$:

$D(0) = 50 e^{-0.55(0)} = 50 e^0 = 50 \text{ milligrams}$

So, the initial amount injected is 50 milligrams.

2. Amount After 6 Hours

To find the amount after 6 hours, substitute $h = 6$ into the equation:

$D(6) = 50 e^{-0.55(6)} = 50 e^{-3.3}$

Using the approximate value of $e^{-3.3}$:

$e^{-3.3} \approx 0.0365$

Now, calculate $D(6)$:

$D(6) = 50 \times 0.0365 = 1.825 \text{ milligrams}$

So, the amount of the drug in the bloodstream after 6 hours is approximately 1.83 milligrams.

Summary

• Initial amount injected: 50 milligrams
• Amount after 6 hours: 1.83 milligrams

Would you like more details on this, or do you have any other questions?

Here are some related questions:

1. How does the decay constant $0.55$ affect the rate at which the drug leaves the bloodstream?
2. What would the amount of the drug be after 12 hours?
3. How long would it take for the drug to reduce to 10% of its initial value?
4. What is the half-life of the drug in this model?
5. How would increasing the initial amount (e.g., from 50 mg to 100 mg) affect the amount after 6 hours?

Tip: Exponential decay functions are commonly used to model processes where the rate of change is proportional to the current amount, such as in pharmacokinetics, radioactive decay, and cooling.