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.

We are given   R(w) = 40 · (w – 3)/(w – 2.5). (a) To find R(4):   R(4) = 40 · (4 – 3)/(4 – 2.5)       = 40 · (1)/(1.5)       = 40/1.5       = 26.67   (rounded to two decimals) (b) To find R(5):   R(5) = 40 · (5 – 3)/(5 – 2.5)       = 40 · (2)/(2.5)       = 80/2.5       = 32.00   (rounded to two decimals) (c) To find the derivative R′(w), we first write   R(w) = 40 · (w – 3)/(w – 2.5). Use the quotient rule for differentiation. Let u(w) = w – 3 and v(w) = w – 2.5 so that u′(w)=1 and v′(w)=1. Then   R′(w) = 40 · [ u′(w)v(w) – u(w)v′(w) ] / [v(w)]²       = 40 · [ (1)(w – 2.5) – (w – 3)(1) ] / (w – 2.5)²       = 40 · [w – 2.5 – w + 3] / (w – 2.5)²       = 40 · (0.5) / (w – 2.5)²       = 20/(w – 2.5)². Now, compute R′(4) and R′(5): For w = 4:   R′(4) = 20/(4 – 2.5)²       = 20/(1.5)²       = 20/2.25       ≈ 8.89   (rounded to two decimals) For w = 5:   R′(5) = 20/(5 – 2.5)²       = 20/(2.5)²       = 20/6.25       = 3.20   (rounded to two decimals) Comparing R′(4) and R′(5):   8.89 (at w = 4) and 3.20 (at w = 5) show that R′(4) ≥ R′(5). Since R′ gives the rate of change of rest minutes with respect to the work rate w, the decrease in the derivative as w increases indicates that the rate of change in the amount of rest required is decreasing. Thus, the correct answer is:   B. R′(4) ≥ R′(5) and the rate of change in the amount of rest required after working for 40 minutes is decreasing as w increases. Final Answers: (a) R(4) = 26.67 minutes (b) R(5) = 32.00 minutes (c) R′(4) ≈ 8.89 min/(kcal/min) and R′(5) ≈ 3.20 min/(kcal/min); Answer: B.