Brenton's weekly pay, [tex]$P(h)$[/tex], in dollars, is a function of the number of hours he works, [tex]$h$[/tex]. He gets paid $20 per hour for the first 40 hours he works in a week. For any hours above that, he is paid overtime at $30 per hour. He is not permitted to work more than 60 hours in a week.
Which set describes the domain of [tex]$P(h)$[/tex]?
A. {h | 0 ≤ h ≤ 40}
B. {h | 0 ≤ h ≤ 60}
C. {P(h) | 0 ≤ P(h) ≤ 1,400}
D. {P(h) | 0 ≤ P(h) ≤ 1,800}
Answer (2)
The domain of the function represents the possible number of hours Brenton can work.
The minimum number of hours is 0, and the maximum is 60.
The domain is the set of all h such that 0 ≤ h ≤ 60 .
Therefore, the correct answer is h ∣ 0 ≤ h ≤ 60 .
Explanation
Understanding the Problem We are given that Brenton's weekly pay, P ( h ) , is a function of the number of hours he works, h . He gets paid $20 per hour for the first 40 hours he works in a week. For any hours above that, he is paid overtime at $30 per hour. He is not permitted to work more than 60 hours in a week. We need to find the domain of P ( h ) . The domain represents the possible values of h .
Determining the Domain The minimum number of hours Brenton can work is 0. The maximum number of hours Brenton can work is 60. Therefore, the domain of P ( h ) is the set of all h such that 0 ≤ h ≤ 60 .
Expressing the Domain The domain of P ( h ) is the set of all possible values of h , which represents the number of hours Brenton can work in a week. Since he can work from 0 to 60 hours, the domain is given by the set hmi d 0 ≤ h ≤ 60 .
Examples
Understanding the domain of a function is crucial in many real-life scenarios. For instance, if you're planning a road trip, the domain could represent the possible number of hours you can drive each day. Knowing this range helps you plan your stops and arrival time effectively. Similarly, in a manufacturing process, the domain might represent the number of units you can produce given certain constraints like available resources or time. Recognizing these limits ensures efficient planning and prevents overproduction or resource depletion.
The domain of Brenton's weekly pay function P ( h ) is based on the number of hours he can work, which ranges from 0 to 60 hours. Thus, the answer is B: {h | 0 \leq h \leq 60}.
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