To solve the problem, we start by interpreting the given information: "The amount of pay, [tex]\( p \)[/tex], that Susan earns varies directly with the number of hours, [tex]\( h \)[/tex], that she works."
When a quantity [tex]\( p \)[/tex] varies directly with another quantity [tex]\( h \)[/tex], it means that [tex]\( p \)[/tex] is directly proportional to [tex]\( h \)[/tex]. In mathematical terms, this can be written as:
[tex]\[ p = k \cdot h \][/tex]
where [tex]\( k \)[/tex] is the constant of variation (or proportionality constant).
Now, let's analyze each of the given options to identify the correct equation:
Option A: [tex]\( p = \frac{k}{h} \)[/tex]
- This suggests that [tex]\( p \)[/tex] is inversely proportional to [tex]\( h \)[/tex], which is not consistent with the statement that [tex]\( p \)[/tex] varies directly with [tex]\( h \)[/tex].
Option B: [tex]\( p = \frac{h}{k} \)[/tex]
- This implies that [tex]\( p \)[/tex] is directly proportional to the reciprocal of [tex]\( k \)[/tex], but it does not conform to the direct variation stated in the problem.
Option C: [tex]\( p = k \cdot h \)[/tex]
- This equation correctly expresses that [tex]\( p \)[/tex] is directly proportional to [tex]\( h \)[/tex] with [tex]\( k \)[/tex] being the proportionality constant or constant of variation. This directly matches the given relationship.
Option D: [tex]\( p = k + h \)[/tex]
- This suggests that [tex]\( p \)[/tex] is the sum of [tex]\( k \)[/tex] and [tex]\( h \)[/tex], which is not indicative of a direct variation relationship between [tex]\( p \)[/tex] and [tex]\( h \)[/tex].
Since the correct representation of the direct variation where [tex]\( p \)[/tex] varies directly with [tex]\( h \)[/tex] is given by [tex]\( p = k \cdot h \)[/tex], the correct equation is:
[tex]\[ p = k h \][/tex]
Thus, the correct answer is:
c. C
