rYzcC0k5 | Mathematico

A rectangle has a width of $k + 2$ and a height of $k - 2,$ where $k$ is a real number.

rectangle

The area of the rectangle is smaller than or equal to its perimeter.

The range of possible values of $k$ is of the form

α < k \leq β

Find the value of $α + β$ to three significant figures.

Hint

The area is

(k + 2) (k - 2)

and the perimeter is

2 (k + 2) + 2 (k - 2)

Hints

Remember that $k - 2$ and $k + 2$ are lengths, so they must be positive.

Solution

The area is smaller than or equal to the perimeter, so

\begin{aligned} (k + 2) (k - 2) & \leq 2 (k + 2) + 2 (k - 2) \\ k^{2} - 4 & \leq 4 k \\ k^{2} - 4 k - 4 \leq 0 \end{aligned}

The roots are

k = 2 \pm 2 \sqrt{2}

We also note that $k - 2$ is a length and therefore must be positive, so we have the additional constraint that $k > 2$ .

We represent this on a sketch:

rectangle

Values of $k$ in the shaded area must be excluded, so we see that the solution is

2 < k \leq 2 + 2 \sqrt{2}