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 0<f(w)<1,050 0 less than f left parenthesis w right parenthesis less than 1,050 −∞<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
Answers
GPT-4o mini
In order to determine the appropriate inequality that represents the range of Dorian's earnings from walking dogs, we can start by analyzing the function given: \( f(w) = 15w \).
### Key Points:
- \( w \) represents the number of dogs he walks.
- Since he cannot walk a negative number of dogs, \( w \) must be non-negative: \( w \geq 0 \).
- The maximum value of \( f(w) \) depends on the context provided regarding Dorian's dog walking activities. If there is no information on an upper limit on the number of dogs he can walk, we consider he could theoretically walk an infinite number, leading to infinite earnings. However, it's more realistic in a typical situation that he might have a limit based on time or availability.
### Analyzing the Options:
1. **\( 0 \leq f(w) < \infty \)**: This represents that his earnings are non-negative and can go to infinity, which is a valid representation as long as there's no limit on the number of dogs he can walk.
2. **\( 0 < f(w) < 1,050 \)**: This means he earns positive amounts but less than $1,050, which implies a specific upper limit on how much Dorian can earn without a basis from the problem's context.
3. **\( 0 < f(w) < 1,050\)**: Same reasoning as above, but with \( f(w) \) not including 0, which doesn't make sense since he could potentially earn nothing if he doesn't walk any dogs.
4. **\( -\infty < f(w) \leq 1,050 \)**: This is incorrect, as f(w) cannot be negative, and also suggests an upper limit which is not provided in the context.
5. **\( 0 \leq f(w) \leq 1,050 \)**: If $1,050 is considered a maximum bound on his earnings, this implies he could now be earning zero or up to $1,050, which is credible if there's a scenario where he's limited to not earning more than $1,050.
Given the provided options and assuming there's no context explicitly stating a limit less than infinity, the appropriate inequality that shows the range of his earnings based on the most typical realistic constraints would be:
### Correct Answer:
- **\( 0 \leq f(w) < \infty \)**
### Key Points:
- \( w \) represents the number of dogs he walks.
- Since he cannot walk a negative number of dogs, \( w \) must be non-negative: \( w \geq 0 \).
- The maximum value of \( f(w) \) depends on the context provided regarding Dorian's dog walking activities. If there is no information on an upper limit on the number of dogs he can walk, we consider he could theoretically walk an infinite number, leading to infinite earnings. However, it's more realistic in a typical situation that he might have a limit based on time or availability.
### Analyzing the Options:
1. **\( 0 \leq f(w) < \infty \)**: This represents that his earnings are non-negative and can go to infinity, which is a valid representation as long as there's no limit on the number of dogs he can walk.
2. **\( 0 < f(w) < 1,050 \)**: This means he earns positive amounts but less than $1,050, which implies a specific upper limit on how much Dorian can earn without a basis from the problem's context.
3. **\( 0 < f(w) < 1,050\)**: Same reasoning as above, but with \( f(w) \) not including 0, which doesn't make sense since he could potentially earn nothing if he doesn't walk any dogs.
4. **\( -\infty < f(w) \leq 1,050 \)**: This is incorrect, as f(w) cannot be negative, and also suggests an upper limit which is not provided in the context.
5. **\( 0 \leq f(w) \leq 1,050 \)**: If $1,050 is considered a maximum bound on his earnings, this implies he could now be earning zero or up to $1,050, which is credible if there's a scenario where he's limited to not earning more than $1,050.
Given the provided options and assuming there's no context explicitly stating a limit less than infinity, the appropriate inequality that shows the range of his earnings based on the most typical realistic constraints would be:
### Correct Answer:
- **\( 0 \leq f(w) < \infty \)**