Part 2 of 2 \( y=597 x^{-1 / 3} \) 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 beats per minute, as given approximately by What is the instantaneous rate of change of pulse rate for a 40 inch tall person? -1.45 beats per minute per inch (Round to the nearest hundredth as needed.) What is the instantaneous rate of change of pulse rate for a 61 inch tall person? \( \square \) beats per minute per inch (Round to the nearest hundredth as needed.)

To find the instantaneous rate of change of pulse rate for a 40-inch tall person, we need to find the derivative of the given function \( y = 597x^{-1/3} \) and then evaluate it at \( x = 40 \). Given function: \( y = 597x^{-1/3} \) To find the derivative of the function, we will use the power rule of differentiation, which states that if \( y = x^n \), then \( y' = nx^{n-1} \). Differentiating the given function: \[ y' = 597 \cdot \left(-\frac{1}{3}\right) \cdot x^{-1/3 - 1} \] \[ y' = -199 \cdot x^{-4/3} \] Now, we need to evaluate the derivative at \( x = 40 \) to find the instantaneous rate of change of pulse rate for a 40-inch tall person: \[ y' = -199 \cdot 40^{-4/3} \] Let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(-199\times 40^{\frac{-4}{3}}\) - step1: Rewrite the fraction: \(-199\times 40^{-\frac{4}{3}}\) - step2: Evaluate the power: \(-199\times \frac{1}{40^{\frac{4}{3}}}\) - step3: Multiply: \(-\frac{199}{40^{\frac{4}{3}}}\) - step4: Simplify: \(-\frac{199}{80\sqrt[3]{5}}\) - step5: Multiply by the Conjugate: \(\frac{-199\sqrt[3]{5^{2}}}{80\sqrt[3]{5}\times \sqrt[3]{5^{2}}}\) - step6: Simplify: \(\frac{-199\sqrt[3]{25}}{80\sqrt[3]{5}\times \sqrt[3]{5^{2}}}\) - step7: Multiply the numbers: \(\frac{-199\sqrt[3]{25}}{400}\) - step8: Calculate: \(-\frac{199\sqrt[3]{25}}{400}\) The instantaneous rate of change of pulse rate for a 40-inch tall person is approximately -1.45 beats per minute per inch. Now, let's find the instantaneous rate of change of pulse rate for a 61-inch tall person by evaluating the derivative at \( x = 61 \): \[ y' = -199 \cdot 61^{-4/3} \] Let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(-199\times 61^{\frac{-4}{3}}\) - step1: Rewrite the fraction: \(-199\times 61^{-\frac{4}{3}}\) - step2: Evaluate the power: \(-199\times \frac{1}{61^{\frac{4}{3}}}\) - step3: Multiply: \(-\frac{199}{61^{\frac{4}{3}}}\) - step4: Simplify: \(-\frac{199}{61\sqrt[3]{61}}\) - step5: Multiply by the Conjugate: \(\frac{-199\sqrt[3]{61^{2}}}{61\sqrt[3]{61}\times \sqrt[3]{61^{2}}}\) - step6: Simplify: \(\frac{-199\sqrt[3]{3721}}{61\sqrt[3]{61}\times \sqrt[3]{61^{2}}}\) - step7: Multiply the numbers: \(\frac{-199\sqrt[3]{3721}}{3721}\) - step8: Calculate: \(-\frac{199\sqrt[3]{3721}}{3721}\) The instantaneous rate of change of pulse rate for a 61-inch tall person is approximately -0.83 beats per minute per inch. Therefore, the answers are: (A) -1.45 beats per minute per inch (B) -0.83 beats per minute per inch