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.)

Question

UpStudy AI · Accepted Answer

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
The instantaneous rate of change of pulse rate for a 40-inch tall person is approximately $$-1.45$$ beats per minute per inch.

Solución de tutoría real

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Calculus Questions