Westworld 1973 Problem #2
As advertised, the Delos theme park offers you their incredible interactive android vacation packages (Western World, Medieval World, & Roman World) for $ a day.
This triptych of Worlds has an estimated capacity for Guests. For every $ increase in the daily price, the number of Guests is estimated to decrease by persons.
Also, the park experiences a robot damage rate equal to of the total Guest population that incurs a $ per robot charge for parts & labor. It also maintains a daily operation cost of for its facilities.
What is the maximum daily revenue, , that can be attained under this Delos pricing model (in dollars)? Enter your answer as
The answer is 2008522.
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1 solution
The daily Revenue takes into account Sales - Costs. On the Sales side, attendance will decrease by 1 0 Guests for every $ 1 0 0 increment to the admission price. This can be expressed as:
S ( x ) = ( 1 0 0 0 − 1 0 x ) ( $ 1 0 0 0 + $ 1 0 0 x ) (i)
On the Costs side, the park has a flat daily operations cost plus a robot repair cost based upon 1 0 % of the total Guest population. This can be expressed as:
C ( x ) = $ 1 0 0 0 0 0 0 + $ 3 0 0 ⋅ 1 0 1 0 0 0 − 1 0 x (ii)
The Revenue is just the difference of (i) and (ii), or R ( x ) = S ( x ) − C ( x ) = ( 1 0 0 0 − 1 0 x ) ( 1 0 0 0 + 1 0 0 x ) − ( 1 0 0 0 0 0 0 + 3 0 0 ⋅ 1 0 1 0 0 0 − 1 0 x ) = − 1 0 0 0 x 2 + 9 0 3 0 0 x − 3 0 0 0 0 (iii).
The first derivative set equal to zero gives:
R ′ ( x ) = 0 ⇒ − 2 0 0 0 x + 9 0 3 0 0 = 0 ⇒ x = 2 0 9 0 3
and the second derivative at this critical point is:
R ′ ′ ( 9 0 3 / 2 0 ) = − 2 0 0 0 < 0 (hence, a global maximum).
Thus, the maximum daily revenue computes to R ( 9 0 3 / 2 0 ) = − $ 1 0 0 0 ( 9 0 3 / 2 0 ) 2 + $ 9 0 3 0 0 ( 9 0 3 / 2 0 ) − $ 3 0 0 0 0 = $ 2 0 0 8 5 2 2 . 5 , or ⌊ R ( 9 0 3 / 2 0 ) ⌋ = $ 2 , 0 0 8 , 5 2 2 .