Question
Dorian earns $15 every time he walks a dog. He models the total amount of money he makes walking dogs with the equation f(w)=15w . Which inequality shows the range of his earnings given the real-world constraints?(1 point) Responses 0<f(w)<1,050 0 less than f left parenthesis w right parenthesis less than 1,050 0≤f(w)<∞ 0 less-than-or-equal-to f left parenthesis w right parenthesis less than infinity 0≤f(w)≤1,050 0 less-than-or-equal-to f left parenthesis w right parenthesis less-than-or-equal-to 1,050 −∞<f(w)≤1,050
Answers
GPT-4o mini
To determine the appropriate inequality showing the range of Dorian's earnings based on the equation \( f(w) = 15w \), we can analyze the situation:
1. **Earnings Structure**: Dorian earns $15 for each dog he walks, meaning:
- If he walks 0 dogs, his earnings \( f(0) = 15 \cdot 0 = 0 \).
- If he walks a certain number \( w \) of dogs, his total earnings are directly proportional to \( w \), and hence they must be non-negative.
2. **Upper Bound**: Since there is a typical constraint that limits the maximum amount of money he can realistically earn, we can assume that there is a cap on his earnings, such as a maximum of $1,050.
Based on this information, the correct inequality representing the range of his earnings should be:
\[
0 \leq f(w) \leq 1,050
\]
This indicates that he can earn from $0 (if he walks zero dogs) to a maximum of $1,050. Therefore, the correct response is:
**0 ≤ f(w) ≤ 1,050**
1. **Earnings Structure**: Dorian earns $15 for each dog he walks, meaning:
- If he walks 0 dogs, his earnings \( f(0) = 15 \cdot 0 = 0 \).
- If he walks a certain number \( w \) of dogs, his total earnings are directly proportional to \( w \), and hence they must be non-negative.
2. **Upper Bound**: Since there is a typical constraint that limits the maximum amount of money he can realistically earn, we can assume that there is a cap on his earnings, such as a maximum of $1,050.
Based on this information, the correct inequality representing the range of his earnings should be:
\[
0 \leq f(w) \leq 1,050
\]
This indicates that he can earn from $0 (if he walks zero dogs) to a maximum of $1,050. Therefore, the correct response is:
**0 ≤ f(w) ≤ 1,050**