Answer:
(a) 140
(b) $1,400
Explanation:
Price of the trip = $1,800 per person
The boat can accommodate 250 passengers.
Since the price of the ticket is reduced by $10 per person if at least 100 passengers sign up.
P(n) = n{1,800 - (n - 100)10}
= n{1,800 - 10n + 1000
P(n) = 2,800n - [tex]10n^{2}[/tex]
We have to maximize P(n) = 800n - [tex]10n^{2}[/tex]
subject to 0≤ n ≤250
P'(n) = 2,800 - 20n
P"(n) = -20
For critical points, solve the equation P'(n) = 0
2,800 - 20n = 0
n = 140
P"(140) = -20 < 0
n = 140 is a point of maxima.
Thus, the number of passenger is 140.
Revenue = P(140)
= 2,800(140) - [tex]10(140)^{2}[/tex]
= 392,000 - 196,000
= $196,000
Therefore,
Price paid by each passenger:
[tex]=\frac{Revenue}{number\ of\ passenger}[/tex]
[tex]=\frac{196,000}{140}[/tex]
= $1,400
