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:

Step1: Factor the quadratic expression

We factor $k^2 + 19k + 60$ by finding two numbers that multiply to 60 and add to 19. These numbers are 15 and 4.
$$k^2 + 19k + 60 = (k+4)(k+15)$$

Step2: Match factors to options

The dimensions of the rectangle are $(k+4)$ and $(k+15)$, both natural numbers since $k$ is a natural number. We compare these to the given options.

Answer:

B. The width of the rectangle is $k + 4$ inches.