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Solution

The print quality is very poor so the answer is on best effort basis.

(a) The charges for a 30 member group , for a trip to the concert are $ 10 per person. Also, for each 1 person above the 30 persons level, the charges for all group members are reduced by $ 0.20 per person. Thus, if there are 35 persons in a group, the charges per person are $ 10 – 5*$ 0.20 = $ 10- $ 1 = $ 9 per person. Then the revenue for the Pearson Travel Inc is $ 35*9 = $ 315.

(b) The Revenue for the Pearson Travel Inc., if x persons are in the group, is R(x) = 300 if x 30 and R(x) = [10- 0.2(x-30)](x) = (10-0.2x+6) x = (16-0.2x)x = -0.2x²+16x   if x > 30.

(c) Since 31*9.8 = 303.8 > 30*10 = 300, it is apparent if the number of persons in the group (i.e. x) > 30. Further, we know that R(x) is maximum when dR/dx = 0 and d²R/dx² is negative. Here, dR/dx = -0.4x +16 and d²R/dx² =-0.4. Thus, if dR/dx = 0, then -0.4x +16 = 0 or, 0.4x = 16 so that x = 16/0.4 = 40. Further, d²R/dx² = -0.4 is negative irrespective of the value of x. Hence the revenue for the Pearson Travel Inc will be maximum when there are 40 persons in the group.