Question
Arturo's math teacher finds that there's roughly a linear relationship between the amount of time students spend on their homework and their weekly quiz scores. This relationship can be represented by the equation y, equals, 56, plus, 8, point, 2, xy=56+8.2x, where yy represents the expected quiz score and xx represents hours spent on homework that week. What is the meaning of the xx-value when y, equals, 89y=89?
Answer
Multiple Choice Answers
A student's expected quiz score if they spent no time on their homework.
A student's expected quiz score if they spent 89 hours on their homework.
The change in expected quiz score for every additional one hour students spend on their homework.
The number of hours a student should spend on their homework to expect a score of 89 on the quiz.
Answer
Multiple Choice Answers
A student's expected quiz score if they spent no time on their homework.
A student's expected quiz score if they spent 89 hours on their homework.
The change in expected quiz score for every additional one hour students spend on their homework.
The number of hours a student should spend on their homework to expect a score of 89 on the quiz.
Answers
GPT-4o mini
To find the meaning of the \( x \)-value when \( y = 89 \) in the equation \( y = 56 + 8.2x \), we need to solve for \( x \) when \( y \) is set to 89:
1. Set the equation to 89:
\[
89 = 56 + 8.2x
\]
2. Subtract 56 from both sides:
\[
89 - 56 = 8.2x
\]
\[
33 = 8.2x
\]
3. Divide both sides by 8.2:
\[
x = \frac{33}{8.2} \approx 4.02
\]
This means that when a student spends approximately 4.02 hours on their homework, their expected quiz score is 89.
Now examining the provided multiple-choice answers:
1. **A student's expected quiz score if they spent no time on their homework.** (This corresponds to \( x = 0 \), which would give \( y = 56 \). This is incorrect.)
2. **A student's expected quiz score if they spent 89 hours on their homework.** (This corresponds to setting \( x = 89 \) in the equation, which does not apply here. This is incorrect.)
3. **The change in expected quiz score for every additional one hour students spend on their homework.** (This corresponds to the slope of the equation, 8.2. This is also incorrect.)
4. **The number of hours a student should spend on their homework to expect a score of 89 on the quiz.** (We calculated that to be approximately 4.02 hours. This is correct.)
Thus, the correct answer is:
**The number of hours a student should spend on their homework to expect a score of 89 on the quiz.**
1. Set the equation to 89:
\[
89 = 56 + 8.2x
\]
2. Subtract 56 from both sides:
\[
89 - 56 = 8.2x
\]
\[
33 = 8.2x
\]
3. Divide both sides by 8.2:
\[
x = \frac{33}{8.2} \approx 4.02
\]
This means that when a student spends approximately 4.02 hours on their homework, their expected quiz score is 89.
Now examining the provided multiple-choice answers:
1. **A student's expected quiz score if they spent no time on their homework.** (This corresponds to \( x = 0 \), which would give \( y = 56 \). This is incorrect.)
2. **A student's expected quiz score if they spent 89 hours on their homework.** (This corresponds to setting \( x = 89 \) in the equation, which does not apply here. This is incorrect.)
3. **The change in expected quiz score for every additional one hour students spend on their homework.** (This corresponds to the slope of the equation, 8.2. This is also incorrect.)
4. **The number of hours a student should spend on their homework to expect a score of 89 on the quiz.** (We calculated that to be approximately 4.02 hours. This is correct.)
Thus, the correct answer is:
**The number of hours a student should spend on their homework to expect a score of 89 on the quiz.**