A formula for the relationship between weight and blood pressure in children is given by the formula below where P(X) is measured in millimeters of mercury and x is measured in pounds. Use the formula to answer the questions. P(X) = 17.2(3+ In x) 10 sxs 100 What is the rate of change of blood pressure with respect to weight at the 50-pound weight level? The rate of change at the 50-pound weight level is approximately mm/pound. (Do not round until the final answer. Then round to the nearest hundredth as needed.)
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![### Educational Exercise: Relationship Between Weight and Blood Pressure in Children
**Problem:** A formula for the relationship between weight and blood pressure in children is given by the formula below:
\[ P(x) = 17.2(3 + \ln x) \]
where \( P(x) \) is measured in millimeters of mercury (mmHg) and \( x \) is measured in pounds. Use the formula to answer the following questions.
**Question:**
1. What is the rate of change of blood pressure with respect to weight at the 50-pound weight level?
The rate of change at the 50-pound weight level is approximately \(\_\_\_\_\_\_\_\_\_\) mm/pound.
(Do not round until the final answer. Then round to the nearest hundredth as needed.)
### Detailed Explanation:
To find the rate of change of blood pressure concerning weight, we need to calculate the derivative of \( P(x) \) with respect to \( x \).
\[
P(x) = 17.2(3 + \ln x)
\]
**Step-by-Step Solution:**
1. Find the derivative \( \frac{dP}{dx} \):
\[
\frac{dP}{dx} = 17.2 \cdot \frac{d}{dx}(3 + \ln x)
\]
\[
\frac{dP}{dx} = 17.2 \cdot \frac{d}{dx}(3) + 17.2 \cdot \frac{d}{dx}(\ln x)
\]
Since the derivative of a constant (3) is 0:
\[
\frac{dP}{dx} = 0 + 17.2 \cdot \frac{1}{x}
\]
\[
\frac{dP}{dx} = 17.2 \cdot \frac{1}{x} = \frac{17.2}{x}
\]
2. Substitute \( x = 50 \) to find the rate of change of blood pressure when the weight \( x \) is 50 pounds:
\[
\frac{dP}{dx} \bigg|_{x=50} = \frac{17.2}{50}
\]
3. Calculate the value:
\[
\frac{17.2}{50} = 0.344
\]
Thus, the rate of change of blood pressure](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5403b22d-6bfd-443b-9939-b04fac993812%2Fccd7046b-5082-4c8e-9bbf-91e9554abb68%2Fz77qz4v_processed.jpeg&w=3840&q=75)
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