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.
Show 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 $x = 1$).
Step 1: Complete the square on the quadratic, instead of using the vertex formula x = -b/2a.
\[ p(x) = -x^2 + 2x + 8 \]
Factor out $-1$ from the first two terms:
\[ p(x) = -(x^2 - 2x) + 8 \]
Step 2: Complete the square inside the brackets.
To complete the square for $x^2 - 2x$, add and subtract $1$ (since half of $-2$ is $-1$, and $(-1)^2 = 1$):
\[ x^2 - 2x = (x^2 - 2x + 1) - 1 = (x-1)^2 - 1 \]
Step 4: Reason about the maximum value directly from this form.
Since $(x-1)^2$ is a square, it can never be negative, so $-(x-1)^2$ can never be positive; its largest possible value is $0$, which happens exactly when $x = 1$.
Therefore, the largest possible value of $p(x)$ is:
\[ p(x)_{\max} = 0 + 9 = 9 \]
occurring at $x = 1$.
Final Answer:
The height of the arch is $9$ feet.
\[ \boxed{9 \text{ feet}} \]
