Certainly! Let's go through the steps to determine the amount of the drug in the bloodstream after 4 hours and after 7 hours using the given function.
The function for the amount of the drug in the bloodstream is:
[tex]\[ D(h) = 40 e^{-0.25 h} \][/tex]
### Step 1: Calculate the amount of the drug after 4 hours
First, plug in [tex]\( h = 4 \)[/tex] into the function:
[tex]\[ D(4) = 40 e^{-0.25 \times 4} \][/tex]
Simplify the exponent:
[tex]\[ D(4) = 40 e^{-1} \][/tex]
Now, calculate the value of [tex]\( e^{-1} \)[/tex]. The mathematical constant [tex]\( e \)[/tex] (approximately 2.71828) raised to the power of -1 is approximately 0.36788.
Therefore:
[tex]\[ D(4) = 40 \times 0.36788 \][/tex]
Multiply these values:
[tex]\[ D(4) \approx 14.72 \][/tex]
So, the amount of the drug in the bloodstream after 4 hours is:
[tex]\[ \boxed{14.72} \text{ milligrams} \][/tex]
### Step 2: Calculate the amount of the drug after 7 hours
Next, plug in [tex]\( h = 7 \)[/tex] into the function:
[tex]\[ D(7) = 40 e^{-0.25 \times 7} \][/tex]
Simplify the exponent:
[tex]\[ D(7) = 40 e^{-1.75} \][/tex]
Now, calculate the value of [tex]\( e^{-1.75} \)[/tex]. The mathematical constant [tex]\( e \)[/tex] raised to the power of -1.75 is approximately 0.16831.
Therefore:
[tex]\[ D(7) = 40 \times 0.16831 \][/tex]
Multiply these values:
[tex]\[ D(7) \approx 6.95 \][/tex]
So, the amount of the drug in the bloodstream after 7 hours is:
[tex]\[ \boxed{6.95} \text{ milligrams} \][/tex]
In summary, the amount of the drug in the bloodstream is approximately:
- After 4 hours: [tex]\( \boxed{14.72} \)[/tex] milligrams
- After 7 hours: [tex]\( \boxed{6.95} \)[/tex] milligrams.
