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)<∞
0 less-than-or-equal-to f left parenthesis w right parenthesis less than infinity
−∞<f(w)≤1,050
negative infinity less than f left parenthesis w right parenthesis less-than-or-equal-to 1,050
0≤f(w)≤1,050
0 less-than-or-equal-to f left parenthesis w right parenthesis less-than-or-equal-to 1,050
0<f(w)<1,050
. Which inequality shows the range of his earnings given the real-world constraints?(1 point)
Responses
0≤f(w)<∞
0 less-than-or-equal-to f left parenthesis w right parenthesis less than infinity
−∞<f(w)≤1,050
negative infinity less than f left parenthesis w right parenthesis less-than-or-equal-to 1,050
0≤f(w)≤1,050
0 less-than-or-equal-to f left parenthesis w right parenthesis less-than-or-equal-to 1,050
0<f(w)<1,050
Answers
GPT-4o mini
To determine the appropriate inequality that reflects Dorian's earnings from walking dogs, we note the following:
- The function \( f(w) = 15w \) indicates that Dorian earns $15 for each dog he walks.
- The number of dogs he can walk (\( w \)) cannot be negative, so \( w \geq 0 \).
- Additionally, the problem suggests there's a maximum earning limit of $1,050.
Let's break down the choices:
1. **\( 0 \leq f(w) < \infty \)**
- This means Dorian earns at least $0 and can earn infinitely, which doesn't fully address the constraint of a maximum earning limit.
2. **\( -\infty < f(w) \leq 1,050 \)**
- This states that Dorian could earn negative amounts, which is not possible since he cannot walk a negative number of dogs.
3. **\( 0 \leq f(w) \leq 1,050 \)**
- This reflects that he can earn $0 (if he walks 0 dogs) and can earn up to $1,050. Given the context, this is valid as \( w \) can be any non-negative integer and the maximum earning corresponds to \( f(w) \) when \( w = 70 \) (since \( 15 \times 70 = 1,050 \)).
4. **\( 0 < f(w) < 1,050 \)**
- This suggests he cannot earn exactly $0 or exactly $1,050, which is limiting and not consistent with the reality of his earnings.
Considering these analyses, the most appropriate inequality that shows the range of Dorian's earnings, given the constraints, is:
**\( 0 \leq f(w) \leq 1,050 \)**.
- The function \( f(w) = 15w \) indicates that Dorian earns $15 for each dog he walks.
- The number of dogs he can walk (\( w \)) cannot be negative, so \( w \geq 0 \).
- Additionally, the problem suggests there's a maximum earning limit of $1,050.
Let's break down the choices:
1. **\( 0 \leq f(w) < \infty \)**
- This means Dorian earns at least $0 and can earn infinitely, which doesn't fully address the constraint of a maximum earning limit.
2. **\( -\infty < f(w) \leq 1,050 \)**
- This states that Dorian could earn negative amounts, which is not possible since he cannot walk a negative number of dogs.
3. **\( 0 \leq f(w) \leq 1,050 \)**
- This reflects that he can earn $0 (if he walks 0 dogs) and can earn up to $1,050. Given the context, this is valid as \( w \) can be any non-negative integer and the maximum earning corresponds to \( f(w) \) when \( w = 70 \) (since \( 15 \times 70 = 1,050 \)).
4. **\( 0 < f(w) < 1,050 \)**
- This suggests he cannot earn exactly $0 or exactly $1,050, which is limiting and not consistent with the reality of his earnings.
Considering these analyses, the most appropriate inequality that shows the range of Dorian's earnings, given the constraints, is:
**\( 0 \leq f(w) \leq 1,050 \)**.