Question

a rectangle has an area of ( k^{2}+14k + 40 ) 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 have the quadratic expression \(k^{2}+14k + 40\).
We need to find two numbers \(m\) and \(n\) such that \(m + n=14\) and \(m\times n = 40\).
The numbers are \(m = 4\) and \(n = 10\) since \(4+10=14\) and \(4\times10 = 40\).
So, \(k^{2}+14k + 40=(k + 4)(k+10)\)

Step2: Recall the area formula for a rectangle

The area of a rectangle \(A=\text{length}\times\text{width}\).
If \(A=(k + 4)(k + 10)\), then the possible dimensions of the rectangle are \((k + 4)\) inches and \((k + 10)\) inches

Answer:

The width of the rectangle is \(k + 10\) inches.