Motley Motel's potential average room rate is calculated to be $62. Assume that this motel had three market segments. Vacation travelers use 75% of the room nights and are charged 100% of the rack rate. Business travelers use 15% of the room nights and are charged 90% of the rack rate. Sports teams account for 10% of the room nights and are charged 80% of the rack rate. 

a. Calculate the room rate by market segment. 

b. Prove that your calculations are correct, assuming that total annual room nights are 7,300.