To determine which function represents the average number of photos taken per passenger as a function of the speed of the ship, we need to follow the relationships outlined in the problem:
1. Duration of the cruise, [tex]\( r \)[/tex]: The duration of the cruise [tex]\( r \)[/tex] is a function of the speed of the ship [tex]\( k \)[/tex]—this relationship is represented by [tex]\( D \)[/tex]. Therefore, the duration of the cruise [tex]\( r \)[/tex] can be expressed as [tex]\( D(k) \)[/tex].
2. Average number of photos taken per passenger, [tex]\( P \)[/tex]: This is a function of the duration of the cruise [tex]\( r \)[/tex], which is expressed as [tex]\( P(r) \)[/tex].
Given that [tex]\( r \)[/tex] is determined by [tex]\( k \)[/tex] through the function [tex]\( D \)[/tex], we need to substitute [tex]\( r \)[/tex] with [tex]\( D(k) \)[/tex] in the function [tex]\( P(r) \)[/tex].
Therefore, the function that represents the average number of photos taken per passenger as a function of the speed of the ship [tex]\( k \)[/tex] is: [tex]\( P(D(k)) \)[/tex].
So the correct answer is:
(A) [tex]\( P(D(k)) \)[/tex]
When referencing the choices given:
- (A) [tex]\( P(D(k)) \)[/tex] correctly represents the average number of photos taken per passenger as a function of the speed of the ship.
- (B) [tex]\( P(S(k)) \)[/tex] is incorrect because [tex]\( S \)[/tex] represents the speed of the ship as a function of the current of the river, not the speed of the ship provided.
- (C) [tex]\( D(P(r)) \)[/tex] and (D) [tex]\( S(P(r)) \)[/tex] are incorrect as they incorrectly relate the functions.
Hence, the answer is:
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