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:

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

$$ \text{The width of the rectangle is } k + 4 \text{ inches.} $$

Answer:

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