Question

a rectangle has an area of \\(k^2 + 19k + 60\\) square inches. if the value of \\(k\\) and the dimensions of the rectangle are all natural numbers, which statement about the rectangle could be true?

• the length of the rectangle is \\(k - 5\\) inches.
• the width of the rectangle is \\(k + 4\\) inches.
• the length of the rectangle is \\(k - 20\\) inches.
• the width of the rectangle is \\(k + 10\\) inches.

Explanation:

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<reasoning>

Factor the quadratic area expression

\[
k^2 + 19k + 60 = (k + 15)(k + 4)
\]

Identify the possible dimensions

\[
\text{Dimensions} = (k + 15) \text{ and } (k + 4)
\]

Match with the given options

\[
\text{The width of the rectangle is } k + 4 \text{ inches.}
\]
</reasoning>

<answer>
<mcq-option>The length of the rectangle is \(k - 5\) inches.</mcq-option>
<mcq-correct>The width of the rectangle is \(k + 4\) inches.</mcq-correct>
<mcq-option>The length of the rectangle is \(k - 20\) inches.</mcq-option>
<mcq-option>The width of the rectangle is \(k + 10\) inches.</mcq-option>
</answer>

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]
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Answer:

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<reasoning>

Factor the quadratic area expression

\[
k^2 + 19k + 60 = (k + 15)(k + 4)
\]

Identify the possible dimensions

\[
\text{Dimensions} = (k + 15) \text{ and } (k + 4)
\]

Match with the given options

\[
\text{The width of the rectangle is } k + 4 \text{ inches.}
\]
</reasoning>

<answer>
<mcq-option>The length of the rectangle is \(k - 5\) inches.</mcq-option>
<mcq-correct>The width of the rectangle is \(k + 4\) inches.</mcq-correct>
<mcq-option>The length of the rectangle is \(k - 20\) inches.</mcq-option>
<mcq-option>The width of the rectangle is \(k + 10\) inches.</mcq-option>
</answer>

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