Question

kate packs snow into 5 identical jars. each jar represents a different depth of snow. kate then lets the snow in each jar completely melt. the table shows the height of the liquid in each jar as it relates to the original depth of snow in the jar.

moisture content of snow
\

$$\begin{tabular}{|c|c|} \\hline snow depth (in.), x & water depth (in.), f(x) \\\\ \\hline 2 & 0.4 \\\\ \\hline 4 & 0.8 \\\\ \\hline 6 & 1.2 \\\\ \\hline 8 & 1.6 \\\\ \\hline 10 & 2.0 \\\\ \\hline \\end{tabular}$$

which statements are true about the relationship between the depth of the snow and the height of water in the jar after the snow is melted? select two options.

the points on a graph representing the relationship lie on a line.
there is 0.4 inch of water to every 1 inch of snow.
a line through the points will pass through (0, 0).
the function relating snow depth to water depth is quadratic.
the data can be represented by f(x) = 0.2^x

Explanation:

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<reasoning>

Analyze the rate of change

\[
\frac{f(x)}{x} = \frac{0.4}{2} = \frac{0.8}{4} = \frac{1.2}{6} = \frac{1.6}{8} = \frac{2.0}{10} = 0.2
\]

Evaluate the statements

• The constant ratio \(0.2\) means the relationship is proportional and linear, passing through \((0, 0)\).
• The slope is \(0.2\), which means there is \(0.2\) inch of water for every \(1\) inch of snow.
• The points lie on a line represented by \(f(x) = 0.2x\).

</reasoning>

<answer>
<mcq-correct>The points on a graph representing the relationship lie on a line.</mcq-correct>
<mcq-option>There is 0.4 inch of water to every 1 inch of snow.</mcq-option>
<mcq-correct>A line through the points will pass through (0, 0).</mcq-correct>
<mcq-option>The function relating snow depth to water depth is quadratic.</mcq-option>
<mcq-option>The data can be represented by \(f(x) = 0.2^x\).</mcq-option>
</answer>

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Answer:

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<reasoning>

Analyze the rate of change

\[
\frac{f(x)}{x} = \frac{0.4}{2} = \frac{0.8}{4} = \frac{1.2}{6} = \frac{1.6}{8} = \frac{2.0}{10} = 0.2
\]

Evaluate the statements

• The constant ratio \(0.2\) means the relationship is proportional and linear, passing through \((0, 0)\).
• The slope is \(0.2\), which means there is \(0.2\) inch of water for every \(1\) inch of snow.
• The points lie on a line represented by \(f(x) = 0.2x\).

</reasoning>

<answer>
<mcq-correct>The points on a graph representing the relationship lie on a line.</mcq-correct>
<mcq-option>There is 0.4 inch of water to every 1 inch of snow.</mcq-option>
<mcq-correct>A line through the points will pass through (0, 0).</mcq-correct>
<mcq-option>The function relating snow depth to water depth is quadratic.</mcq-option>
<mcq-option>The data can be represented by \(f(x) = 0.2^x\).</mcq-option>
</answer>

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