A person \( x \) inches tall has a pulse rate of \( y \) beats per minute, as given approximately by \( y=597 x^{-1 / 3} \) for \( 30 \leq x \leq 75 \). What is the instantaneous rate of change of pulse rate for the following heights? (A) 40 -inches (B) 61 -inches What is the instantaneous rate of change of pulse rate for a 40 inch tall person? \( \square \) beats per minute per inch (Round to the nearest hundredth as needed.)

To find the instantaneous rate of change of the pulse rate \( y \) with respect to height \( x \) for a 40-inch tall person, we'll follow these steps: 1. **Given Function:** \[ y = 597x^{-\frac{1}{3}} \] 2. **Compute the Derivative \( \frac{dy}{dx} \):** \[ \frac{dy}{dx} = 597 \cdot \left(-\frac{1}{3}\right) x^{-\frac{4}{3}} = -199x^{-\frac{4}{3}} \] 3. **Evaluate the Derivative at \( x = 40 \):** \[ \left.\frac{dy}{dx}\right|_{x=40} = -199 \cdot \frac{1}{40^{\frac{4}{3}}} \] 4. **Calculate \( 40^{\frac{4}{3}} \):** \[ 40^{\frac{1}{3}} \approx 3.42 \quad \text{(since \( 3.42^3 \approx 40 \))} \] \[ 40^{\frac{4}{3}} = (40^{\frac{1}{3}})^4 \approx 3.42^4 \approx 136.80 \] 5. **Final Calculation:** \[ \left.\frac{dy}{dx}\right|_{x=40} \approx -199 \div 136.80 \approx -1.45 \ \text{beats per minute per inch} \] **Answer:** \(-1.45\) beats per minute per inch