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.(1 point)
The domain can be represented by the inequality
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.(1 point)
The domain can be represented by the inequality
Answers
GPT-4o mini
To determine the domain of the function \( y = -1.2x + 120 \), we need to consider the constraints imposed by the real-world context of Lowell's running routine.
1. **Running Days**: Since Lowell runs daily until he reaches his goal of 120 miles, \( x \) (the number of days he runs) must be a non-negative integer. Thus, the minimum value for \( x \) is 0.
2. **Goal Constraint**: To find the maximum number of days he can run, we set \( y \) (the distance remaining to run) to 0, as he stops running once he reaches his goal. Therefore, we solve the equation \( 0 = -1.2x + 120 \):
\[
1.2x = 120
\]
\[
x = \frac{120}{1.2} = 100
\]
This means that Lowell will run for a maximum of 100 days.
Putting these constraints together, the domain of the function can be expressed as:
\[
0 \leq x \leq 100
\]
Thus, the domain can be represented by the inequality:
\[
0 \leq x \leq 100
\]
1. **Running Days**: Since Lowell runs daily until he reaches his goal of 120 miles, \( x \) (the number of days he runs) must be a non-negative integer. Thus, the minimum value for \( x \) is 0.
2. **Goal Constraint**: To find the maximum number of days he can run, we set \( y \) (the distance remaining to run) to 0, as he stops running once he reaches his goal. Therefore, we solve the equation \( 0 = -1.2x + 120 \):
\[
1.2x = 120
\]
\[
x = \frac{120}{1.2} = 100
\]
This means that Lowell will run for a maximum of 100 days.
Putting these constraints together, the domain of the function can be expressed as:
\[
0 \leq x \leq 100
\]
Thus, the domain can be represented by the inequality:
\[
0 \leq x \leq 100
\]