Question

a lake near susans house is predicted to lose water every day because of a drought. the table shows the estimated water depth in feet, \\(y\\), after \\(x\\) days of drought.

choose the function that represents the data.

\\(y = -0.5x + 100\\)
\\(y = -x^{0.5} + 100\\)
\\(y = -0.5x^2 + 100\\)
\\(y = -\frac{x}{0.5} + 100\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Linear Functions",
"Rate of Change"
],
"new_concepts": [],
"current_concepts": [
"Linear Functions",
"Rate of Change",
"Linear Modeling"
]
}
</pre_analysis>

<reasoning>

Calculate the rate of change

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{99.5 - 100}{1 - 0} = -0.5
\]

Identify the y-intercept

\[
b = y(0) = 100
\]

Formulate the linear equation

\[
y = mx + b \implies y = -0.5x + 100
\]
</reasoning>

<answer>
<mcq-correct>(A) \(y = -0.5x + 100\)</mcq-correct>
<mcq-option>(B) \(y = -x^{0.5} + 100\)</mcq-option>
<mcq-option>(C) \(y = -0.5x^2 + 100\)</mcq-option>
<mcq-option>(D) \(y = -\frac{x}{0.5} + 100\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Linear Modeling"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Linear Functions",
"Rate of Change"
],
"new_concepts": [],
"current_concepts": [
"Linear Functions",
"Rate of Change",
"Linear Modeling"
]
}
</pre_analysis>

<reasoning>

Calculate the rate of change

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{99.5 - 100}{1 - 0} = -0.5
\]

Identify the y-intercept

\[
b = y(0) = 100
\]

Formulate the linear equation

\[
y = mx + b \implies y = -0.5x + 100
\]
</reasoning>

<answer>
<mcq-correct>(A) \(y = -0.5x + 100\)</mcq-correct>
<mcq-option>(B) \(y = -x^{0.5} + 100\)</mcq-option>
<mcq-option>(C) \(y = -0.5x^2 + 100\)</mcq-option>
<mcq-option>(D) \(y = -\frac{x}{0.5} + 100\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Linear Modeling"
]
}
</post_analysis>