Question

a rectangle has an area of k² + 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:

Step1: Factor the quadratic expression

We factor \(k^{2}+19k + 60\). We need to find two numbers that multiply to \(60\) and add up to \(19\). The numbers are \(15\) and \(4\). So, \(k^{2}+19k + 60=(k + 15)(k+4)\).

Step2: Analyze the factors as dimensions

Since the area of a rectangle \(A = l\times w\) and \(A=k^{2}+19k + 60=(k + 15)(k + 4)\), and \(k\) is a natural - number, the possible length and width of the rectangle are \((k + 15)\) inches and \((k + 4)\) inches.

Answer:

The width of the rectangle is \(k + 4\) inches.