The formula for the total amount of rest, in minutes, required after performing a particular type of work for 40 minutes is given by the formula \( R(w)=40 \frac{w-3}{w-2.5} \), where \( w \) is the work expended in kilocalories per minute (kcal/min). Answer parts (a)-(c). (a) A value of 4 for w indicates light work, such as riding a bicycle on a flat surface at 10 miles per hour. Find \( R(4) \). \( R(4)=\square \) minutes (Type an integer or decimal rounded to two decimal places as needed.) (b) A value of 5 for w indicates moderate work, such as mowing grass with a push mower. Find \( R(5) \). \( R(5)=\square \) minutes (Type an integer or decimal rounded to two decimal places as needed.) (c) Find \( R^{\prime}(4) \) and \( R^{\prime}(5) \). \( R^{\prime}(4)=\square \) min/(kcal/min) and \( R^{\prime}(5)=\square \) min/(kcal/min). (Type integers or decimals rounded to two decimal places as needed.) Compare your answers for \( R^{\prime}(4) \) and \( R^{\prime}(5) \). Choose the correct answer. A. \( R^{\prime}(4) \geq R^{\prime}(5) \) and the rate of change in the amount of rest required after working for 40 minutes is remaining unchanged. B. \( R^{\prime}(4) \geq R^{\prime}(5) \) and the rate of change in the amount of rest required after working for 40 minutes is decreasing as w increases. C. \( R^{\prime}(4) \leq R^{\prime}(5) \) and the rate of change in the amount of rest required after working for 40 minutes cannot be determined. D. \( R^{\prime}(4) \leq R^{\prime}(5) \) and the rate of change in the amount of rest required after working for 40 minutes is decreasing as w increases.

To compute \( R(4) \) and \( R(5) \), we can substitute w in the formula provided. (a) \[ R(4) = 40 \frac{4-3}{4-2.5} = 40 \frac{1}{1.5} = 40 \times \frac{2}{3} \approx 26.67 \text{ minutes}\] (b) \[ R(5) = 40 \frac{5-3}{5-2.5} = 40 \frac{2}{2.5} = 40 \times \frac{4}{5} = 32 \text{ minutes}\] Now, to find the derivatives \( R'(w) \): \[ R(w) = 40 \frac{w-3}{w-2.5} \] Using the quotient rule for derivatives: \[ R'(w) = \frac{(w-2.5)(40) - (w-3)(40)}{(w-2.5)^2} \] Calculating at \( w=4 \) and \( w=5 \): (c) After applying the derivative and plugging in values: \[ R'(4) \approx 53.33 \text{ min/(kcal/min)}\] \[ R'(5) \approx 40 \text{ min/(kcal/min)}\] Comparing \( R'(4) \) and \( R'(5) \): Since \( R'(4) \geq R'(5) \) and the amount of rest required is decreasing as work \( w \) increases, the correct answer is: B. \( R^{\prime}(4) \geq R^{\prime}(5) \) and the rate of change in the amount of rest required after working for 40 minutes is decreasing as \( w \) increases. On to some more engaging facts to consider! Did you know that the relationship between work and rest is something that's been studied for centuries? Ancient Greeks believed in the importance of balancing exertion with downtime, a concept embodied in the saying, “All work and no play makes Jack a dull boy.” Wondering how to optimize your rest and recovery after physical activities? Don’t forget to hydrate and incorporate light stretching or foam rolling to reduce muscle tension! Avoid common mistakes like skipping rest days or not refueling your body after exercise, as these can lead to burnout or injury. Happy resting!