Final answer:
a) To determine the profitable years, solve the inequality S'(x)(1 - C'(x)) > 0. Find x values where C'(x) < 1. Solve the equation x⁴ + (15/4)x³ - 150 = 0. Approximate solutions are x ≈ 6.36 and x ≈ -8.99. Profitable years ≈ 6. b) Substitute x = 1 into S'(x) and C'(x)S'(x) = 150 - x² to find net total savings during the first year. So, the answer is 105.75 dollars. c) Calculate the net total savings over the entire period by integrating the difference between the rate of annual savings and the rate of annual costs. Substitute x = 6 and evaluate the integral to get 657 dollars.
Step-by-step explanation:
a) To determine how many years it will be profitable to use the new machine, we need to find the years where the net total savings are positive. The net total savings is given by the difference between the rate of annual savings and the rate of annual costs. So, we need to solve the inequality S'(x) - C'(x)S'(x) > 0. Simplifying this expression, we get S'(x)(1 - C'(x)) > 0. Solving the inequality 1 - C'(x) > 0, we find that C'(x) < 1. Now, let's analyze the equation 150 - x² = x² + 11/4x. Simplifying this equation, we get x⁴ + (15/4)x³ - 150 = 0. Using a graphing calculator or numerical methods, we find that the solutions are approximately x ≈ 6.36 and x ≈ -8.99. Since the number of years cannot be negative, we ignore the negative solution. Therefore, it will be profitable to use the new machine for approximately 6 years.
b) The net total savings during the first year of use of the machine can be found by calculating the difference between the rate of annual savings and the rate of annual costs at x = 1. Substitute x = 1 into S'(x) and C'(x)S'(x) = 150 - x², we get S'(1) = 150 - 1 = 149 and C'(1)S'(1) = (1)² + 11/4(1)(149) = 43.25. Therefore, the net total savings during the first year of use of the machine is 149 - 43.25 = 105.75 dollars.
c) The net total savings over the entire period of use of the machine can be calculated by integrating the difference between the rate of annual savings and the rate of annual costs over the range of x from 0 to the number of years of operation of the machine. So, the net total savings is given by the integral ∫[0, x] (S'(t) - C'(t)S'(t)) dt. Evaluating this integral, we get ∫[0, x] (150 - t² - t² - 11/4t) dt = ∫[0, x] (150 - 2t² - 11/4t) dt = 150t - 2/3t³ - 11/8t² evaluated from 0 to x. Simplifying further, we get 150x - 2/3x³ - 11/8x² - (0 - 0) = 150x - 2/3x³ - 11/8x². Substituting x = 6, we get 150(6) - 2/3(6)³ - 11/8(6)² = 900 - 144 - 99 = 657 dollars. Therefore, the net total savings over the entire period of use of the machine is 657 dollars.