Question

Case Study - 1: During a theatre drama, a backdrop of building arches was used. The shape of the curve shown below can be represented by the polynomial $p(x) = -x^2 + 2x + 8$, where x is the length (in feet) on stage level. Based on the figure given, answer the following questions :

(i) Determine the height of the arch.

Hint:

Completing the square is a great way to double check:
\[ p(x) = -(x^2 - 2x) + 8 = -(x^2 - 2x + 1 - 1) + 8 = -(x-1)^2 + 9 \] Since the squared term is always positive or zero, the maximum value of the expression is $9$ (which happens when $x = 1$).

Answer 1

Step 1: Understanding the Question:
This case study is from "Polynomials" and "Quadratic Equations".
The curve representing the stage arch is a downward-opening parabola given by $p(x) = -x^2 + 2x + 8$.
The height of the arch corresponds to the maximum vertical value (vertex) of this parabola.

Step 2: Key Formula or Approach:
We can find the maximum value of a quadratic polynomial $ax^2 + bx + c$ by:
1. Finding the x-coordinate of the vertex using:
\[ x = -\frac{b}{2a} \] 2. Substituting this x-value back into $p(x)$ to find the maximum height.
Alternatively, we can rewrite the polynomial in the vertex form by completing the square:
\[ p(x) = a(x-h)^2 + k \] where $k$ is the maximum value.

Step 3: Detailed Explanation:

• Write down the polynomial:
\[ p(x) = -x^2 + 2x + 8 \] Identify coefficients: $a = -1$, $b = 2$, $c = 8$.

• Find the x-coordinate of the vertex:
\[ x = -\frac{2}{2(-1)} = 1 \]

• Substitute $x = 1$ into $p(x)$ to find the maximum height ($k$):
\[ p(1) = -(1)^2 + 2(1) + 8 \] \[ p(1) = -1 + 2 + 8 = 9 \] So the maximum height of the arch is 9 feet.


Step 4: Final Answer:
The height of the arch is $9$ feet.