A certain drug raises a patient's heart rate, h, in beats per minute, according to the equation `h(x)=65+0.2x`, where x is the number of milligrams of the drug in the patient's bloodstream. After t hours, the level of the drug in the patient's bloodsteam decrease exponentially according to the equation `x(t)=512(0.7)^(t)`. After 5 hours, what is the number of beats per minute in the patient's heart rate, correct to the nearest whole number?

Answer

To solve the problem step-by-step, we will follow the equations provided and perform the necessary calculations.

### Step 1: Determine the amount of the drug in the bloodstream after 5 hours
We start with the equation for the amount of the drug in the bloodstream after time \( t \):
\[ x(t) = 512(0.7)^t \]
We need to find \( x(5) \):
\[ x(5) = 512(0.7)^5 \]

### Step 2: Calculate \( (0.7)^5 \)
Using a calculator, we find:
\[ (0.7)^5 \approx 0.16807 \]

### Step 3: Substitute back to find \( x(5) \)
Now we substitute this value back into the equation:
\[ x(5) = 512 \times 0.16807 \approx 86.05 \]

### Step 4: Determine the heart rate using the amount of the drug
Now that we have \( x(5) \), we can find the heart rate \( h(x) \) using the equation:
\[ h(x) = 65 + 0.2x \]
Substituting \( x = 86.05 \):
\[ h(86.05) = 65 + 0.2 \times 86.05 \]

### Step 5: Calculate \( 0.2 \times 86.05 \)
Calculating this gives:
\[ 0.2 \times 86.05 = 17.21 \]

### Step 6: Add to 65 to find the heart rate
Now we can find the heart rate:
\[ h(86.05) = 65 + 17.21 = 82.21 \]

### Step 7: Round to the nearest whole number
Finally, rounding \( 82.21 \) to the nearest whole number gives:
\[ h(86.05) \approx 82 \]

### Final Answer
The number of beats per minute in the patient's heart rate after 5 hours is approximately **82 beats per minute**.

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