Question
Lowell runs 1.2 miles daily. His goal is to run 120 miles, and he will end his daily runs when he reaches that goal. He uses the equation y=−1.2x+120
to keep track of the distance he still needs to run over time. In this function, x represents the number of days he runs, and y represents the number of miles he must still run to achieve his goal. Complete the inequality to determine the domain of the function given the real-world constraints of the problem.
to keep track of the distance he still needs to run over time. In this function, x represents the number of days he runs, and y represents the number of miles he must still run to achieve his goal. Complete the inequality to determine the domain of the function given the real-world constraints of the problem.
Answers
GPT-4o mini
To determine the domain of the function based on the real-world constraints, we need to consider the context of the problem.
1. **Understanding the Running Context**:
- Lowell runs 1.2 miles each day until he reaches a total of 120 miles.
- He will stop running once he has achieved his goal of 120 miles.
2. **Defining x**:
- The variable \( x \) represents the number of days he runs.
- Since he is running a positive distance each day, \( x \) must be a non-negative integer (0 or positive).
3. **Finding the Upper Bound for x**:
- To find how many days it takes to reach his goal:
\[
\text{Total miles run} = 1.2x = 120
\]
Solving for \( x \):
\[
x = \frac{120}{1.2} = 100
\]
- This means Lowell will run for a maximum of 100 days to reach his goal.
4. **Setting Up the Inequality**:
- From the above reasoning, the values for \( x \) must satisfy:
- \( x \geq 0 \) (he cannot run for a negative number of days)
- \( x \leq 100 \) (he stops after reaching 120 miles)
5. **Complete the Domain**:
- Therefore, the proper domain of the function is:
\[
0 \leq x \leq 100
\]
In interval notation, the domain of the function is \( [0, 100] \).
1. **Understanding the Running Context**:
- Lowell runs 1.2 miles each day until he reaches a total of 120 miles.
- He will stop running once he has achieved his goal of 120 miles.
2. **Defining x**:
- The variable \( x \) represents the number of days he runs.
- Since he is running a positive distance each day, \( x \) must be a non-negative integer (0 or positive).
3. **Finding the Upper Bound for x**:
- To find how many days it takes to reach his goal:
\[
\text{Total miles run} = 1.2x = 120
\]
Solving for \( x \):
\[
x = \frac{120}{1.2} = 100
\]
- This means Lowell will run for a maximum of 100 days to reach his goal.
4. **Setting Up the Inequality**:
- From the above reasoning, the values for \( x \) must satisfy:
- \( x \geq 0 \) (he cannot run for a negative number of days)
- \( x \leq 100 \) (he stops after reaching 120 miles)
5. **Complete the Domain**:
- Therefore, the proper domain of the function is:
\[
0 \leq x \leq 100
\]
In interval notation, the domain of the function is \( [0, 100] \).