The Mile High Travel MHT agency offers tours of downtown Den

The Mile High Travel (MHT) agency offers tours of downtown Denver for groups of size 25 people or more. If the group contains exactly 25 people, MHT charges a ticket price of dollar 305 per person. However, the price per person is reduced by dollar 8 for each additional person above the 25. Their tour bus holds a maximum of 45 passengers (not including the bus driver). We\'ll want to maximize revenue, so we\'ll step through this problem in order to figure out the maximum revenue. We have no variables defined in the statement of the problem! So, we\'ll need to define one. The typical way to assign a variable for this type of problem follows. Let x be the number of people who take the tour tAicgalhe minimum 25. Write an equation for the price p(x) of a ticket when x people more than 25 are taking the tour. Set up a revenue function R(x) that gives the total revenue Mile High Travel will receive when they have x more than 25 people on the tour. Figure out the \"reasonable domain

Solution

a. Let the number of persons who take the tour be 25 + x. Then the price of the ticket p (x) = 305 - 8x.

b.The revenue R (x) = number of persons * ticket price = ( 25 + x) (305 - 8x) = 7625 + 105x - 8x² ( however x 20 as the capacity of the tour bus is 45).

c. If R (x) is to be maximum, the dR/dx = 0 and d²R/dx² should be negative. Here, dR/dx = 105- 16x. Thus if dR/dx = 0, then 105 -16x = 0 or, 16x = 105 so that x = 6.5625 = 7 ( on rounding off to the nearest whole number; howewver, for our satisfation, we woill check the value of R ( x) when x = 6 also). Also, d²R/dx² = -16. Thus R (x ) is maximum when x = 7. Now, when x = 7, r (x) = (25 + 7) ( 305 -8*7) = 32( 305 - 56) = 32*249 = 7968. Also, when x = 6, R(x) = (25 +6)( 305 -8 *6) = 31* 257 = 7967. Thus, the maximum revenue from one tour is 7968.

d. The number of people on the maximum revenue tour will be 25 + 7 = 32 ( we may observe that 32 < 45).

3.The ticket price charged per person will be 305 - 8*7 = 305 - 56 = 249.