Question

$ Given the revenue and cost functions R = 36% - 0.6x? and C - 6x + 9, where x is the daily production, find the rate of change of profit with respect to time when 20 units are produced and the rate of change of production is 6 units per day O A $156 per day OB. $72 per day OC. $172.8 per day OD. $36 per day

Answer 1

According to the question, revenue function is, R=36x - 0.6x² The the be and cost function is c = 6x + 9 2 Now, let, P the profit function. By definition of the profit function, we get, P = R-C P = (36% - 0.60²) - (6x + 9) [By ① and ② P = 36x-0.6x² - 6x9 P= – 0.6x² + 362 - 6x – 9 P = -0.6x² + 30x – 9. So, the profit function is, P= – 0.6 x² + 30x - 9. 3 get the rate of change of profit with to time by differentiating 'P' with respect to time. Now, differentiating 'P' with respect to time 'ť get. dP d (-0.6x² + 30x - -g) [By ③ dt dP -0.6. d t (x²) + 30 db dt d (x²)= dP we respect dt d (9) dt .. = 2a da dt dh + 30 da = -0.6x2 ada dt at dt dt da dt d# (x) (9) = 0 dt => dp -1.2 ada dt dt + 30 da dt

is, Thus, the rate of change of profit with respect to time de of 1.2 x dt at dt da 30 da 30 da 2 is the dt the with respect where, daily production and da rate of change of production respect to time. Now, according to the question, we have find the value of dp when dt 20 units and det 6 units per day. dt So, putting and da in 6. we get, 2= 2 = 20 at dP dt = -1.2 x 20 x 6 + 30x6 11 6 (30 - 1.2 x 20) = 6 (30-24) = 6x6 = 36 r = 20 Thus, the value of dp when dt and da is 36. dt Hence, the rate of change of profit with respect to time when 20 units are produced and the rate of change of production is 6 urrits per day is $36 per day. Hence, the answer OD. $36 per day

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