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Question
A tour operator charges ₹ 136 per passenger with a discount of 40 paise for each passenger in excess of 100. The operator requires at least 100 passengers to operate the tour. Determine the number of passengers that will maximize the amount of money the tour operator receives.
Solution
Let x be the required number of passengers
Tour operator charges
`= 136 - 40/100 (x - 100)`, for x ≥ 100
`= 136 - (4x)/10 + 4/10 xx 100`
`= 136 - (4x)/10 + 40`
`= 176 - (2x)/5`
Amount of money, A = (Number of passengers) × (Tour operator charges)
A = `x(176 - (2x)/5)`
A = `176x - (2x)^2/5`
`"dA"/"dx" = 176 - (4x)/5`
When `"dA"/"dx"` = 0 we get,
`176 - (4x)/5 = 0`
`176 = (4x)/5`
4x = 176 × 5
x = `(176 xx 5)/4` = 220
`("d"^2"A")/"dx"^2 = - 4/5`, negative
∴ The amount of money is maximum when the number of passengers is 220.
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