Questions: A person x inches tall has a pulse rate of y beats per minute, as given approximately by y=576 x^(-1 / 3) for 30 ≤ x ≤ 75. What is the instantaneous rate of change of pulse rate for the following heights? (A) 47-inches (B) 71-inches What is the instantaneous rate of change of pulse rate for a 47 inch tall person? beats per minute per inch (Round to the nearest hundredth as needed.) What is the instantaneous rate of change of pulse rate for a 71 inch tall person? beats per minute per inch (Round to the nearest hundredth as needed.)

Solution Steps

To find the instantaneous rate of change of the pulse rate \( y \) with respect to height \( x \), we need to compute the derivative of the given function \( y = 576 x^{-1/3} \). Once we have the derivative, we can evaluate it at the given heights (47 inches and 71 inches).

1. Compute the derivative of \( y \) with respect to \( x \).
2. Evaluate the derivative at \( x = 47 \) inches.
3. Evaluate the derivative at \( x = 71 \) inches.

We start with the function for pulse rate given by

\[ y = 576 x^{-1/3} \]

To find the instantaneous rate of change of pulse rate with respect to height, we compute the derivative \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = -192 x^{-4/3} \]

Next, we evaluate the derivative at \( x = 47 \):

\[ \frac{dy}{dx} \bigg|_{x=47} = -192 \cdot 47^{-4/3} \approx -1.1320 \]

Rounding to the nearest hundredth, we have:

\[ \frac{dy}{dx} \bigg|_{x=47} \approx -1.13 \text{ beats per minute per inch} \]

Now, we evaluate the derivative at \( x = 71 \):

\[ \frac{dy}{dx} \bigg|_{x=71} = -192 \cdot 71^{-4/3} \approx -0.6531 \]

Rounding to the nearest hundredth, we find:

\[ \frac{dy}{dx} \bigg|_{x=71} \approx -0.65 \text{ beats per minute per inch} \]

Final Answer