A small cruising ship that can hold up to 68 people provides

A small cruising ship that can hold up to 68 people provides three-day excursions to groups of 46 or more. If the group contains 46 people, each person pays $70. The cost per person for all members of the party is reduced by $1 for each person in excess of 46. Find the size of the group that maximizes income for the owners of the ship.

Solution

let number of extra members =x

total number of members =46+x

each person pays= 70-x $

toal income i =(46+x)(70-x)

i =3220+24x -x²

for maximum income di/dx =0 ,d²i/dx²<0

di/dx =0+24 -2x=0

=>x=12

d²i/dx²=-2 <0

size of group that may maximises income =46+12=58

income with 46 members =46*70=3220

income with 58 members =58*58=3364

income with 68 members =68*48=3264

size of group that maximises income =58