https://file.aiquickdraw.com/mathbot/file/1728238525267us5j4lyf.jpg

The given formula for the relationship between weight \( x \) (in pounds) and blood pressure \( P(x) \) (in mmHg) is:

\[
P(x) = 19.9(4 + \ln x)
\]

The problem asks for the rate of change of blood pressure with respect to weight, which is the derivative of \( P(x) \) with respect to \( x \). Let’s first find the derivative \( P'(x) \):

\[
P'(x) = \frac{d}{dx} \left[ 19.9(4 + \ln x) \right] = 19.9 \cdot \frac{d}{dx} \left[ 4 + \ln x \right] = 19.9 \cdot \frac{1}{x}
\]

Thus, the derivative is:

\[
P'(x) = \frac{19.9}{x}
\]

### 1. Rate of Change at \( x = 30 \):
Substitute \( x = 30 \) into the derivative:

\[
P'(30) = \frac{19.9}{30} = 0.6633 \text{ mmHg/pound}
\]

Rounding to the nearest hundredth:

\[
P'(30) \approx 0.66 \text{ mmHg/pound}
\]

This matches the provided answer for the 30-pound weight level.

### 2. Rate of Change at \( x = 40 \):
Substitute \( x = 40 \) into the derivative:

\[
P'(40) = \frac{19.9}{40} = 0.4975 \text{ mmHg/pound}
\]

Rounding to the nearest hundredth:

\[
P'(40) \approx 0.50 \text{ mmHg/pound}
\]

So, the rate of change at the 40-pound weight level is approximately **0.50 mmHg/pound**.

Would you like further details or explanations on this? Here are five related questions:

1. How does the natural logarithm \( \ln x \) affect the rate of change as \( x \) increases?
2. Can we approximate the blood pressure at \( x = 50 \) using this formula?
3. What happens to the rate of change as \( x \) approaches the upper limit of 100 pounds?
4. How does the formula's behavior change if the constant 19.9 is modified?
5. How would you compute the total increase in blood pressure between 30 and 40 pounds?

**Tip:** When calculating derivatives involving logarithms, remember that the derivative of \( \ln x \) is always \( \frac{1}{x} \).

What is the rate of change of blood pressure with respect to weight at the 30-pound and 40-pound weight levels?

Learn how to calculate the rate of change of blood pressure with respect to weight using the formula P(x) = 19.9(4 + ln x). This problem involves finding derivatives and logarithmic functions. We calculate the rate of change at the 30-pound and 40-pound weight levels. Suitable for students in Grades 10-12 studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation