Mathematico | Mathematico

The curves shown have equations

\begin{aligned} y & = x^{2} - 8 x + 16 \\ y & = 8 x - x^{2} - 12 \end{aligned}

The area of the depicted rectangle can be expressed in the form $2^{k}$

Find the value of $k$ as an exact decimal.

Hint

To find the height of the rectangle, we need to find the vertex of the concave parabola.

Hint

To find the base, we need the $x$ coordinates of the points of intersection

x^{2} - 8 x + 16 = 8 x - x^{2} - 12

Solution

To find the height of the rectangle, we need the vertex of the concave parabola. We can find it by completing the square

\begin{aligned} y & = - [x^{2} - 8 x] - 12 \\ = - [{(x - 4)}^{2} - 16] - 12 \\ = 4 - (x - 4)^{2} \end{aligned}

So the vertex is at $(4, 4)$ , and the height of the rectangle is $4$ .

For the width of the rectangle, we need the points of intersection between the parabolas:

\begin{aligned} x^{2} - 8 x + 16 & = 8 x - x^{2} - 12 \\ 2 x^{2} - 16 x + 28 & = 0 \\ x^{2} - 8 x + 14 & = 0 \\ (x - 4)^{2} & = 2 \\ x & = 4 \pm \sqrt{2} \end{aligned}

So the width of the rectangle is

(4 + \sqrt{2}) - (4 - \sqrt{2}) = 2 \sqrt{2}

The area of the rectangle is therefore

4 \times 2 \sqrt{2} = 2^{\frac{7}{2}}