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Optimization Problems

The Eco-venture charter company offers local environmental excursions. The fare is $45 per person if up to 30 passengers sign up for the trip. (Note: The company does not offer the trip if fewer than 18 people sign up.) If more than 30 people sign up, the fare for every passenger is reduced by $1 for each passenger in excess of 30. The bus can hold only 48 passengers. Determine the number of passengers that generates the greatest revenue for the charter company. Show all your work. 

Comments

Since you don't offer any cost than it behooves the company to fill every seat regardless of fare.
Behooves is a great word.  Nice work Deanna L.  Sometimes those story questions become challenging, even for adults.   Peace. Samuel
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1 Answer

Let the number of passengers = n + 30 where n = the number of passengers above 30
 
Revenue = (n+30)($45-n) = 45n - n2 + 1350 - 30n = -n2 + 15n + 1350
 
Take the derivative of Revenue wrt n, set it to zero, solve for n:
 
dRevenue/dn = -2n + 15
 
0 = -2n + 15
 
n = 7.5
 
Number of passengers = 30 + 7.5 = 37.5 ≈ 37 or 38 passengers (both generate $1406 in revenue).
 
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You could also solve it algebraically.  The Revenue equation is a quadratic.  Since the coefficient of the n2 term is negative, it's graph is an inverted parabola, so the vertex is the maximum point.  The n -coordinate of the vertex is -b/2a = -15/2(-1) = 7.5.