Let's solve this problem step-by-step in detail:
1. Starting with the given quadratic function:
[tex]\[ f(x) = -4x^2 + 24x - 29 \][/tex]
2. We aim to write this equation in vertex form, which is:
[tex]\[ f(x) = a(x-h)^2 + k \][/tex]
3. First, we need to factor out the coefficient of \(x^2\) from the first two terms:
[tex]\[ f(x) = -4(x^2 - \frac{24}{-4}x) - 29 \][/tex]
[tex]\[ f(x) = -4(x^2 - 6x) - 29 \][/tex]
4. Next, we complete the square inside the parentheses:
To complete the square:
- Take half the coefficient of \(x\), which is \(-6\), and square it.
Half of \(-6\) is \(-3\), and \((-3)^2\) is \(9\).
5. Add and subtract this square inside the parentheses:
[tex]\[ f(x) = -4(x^2 - 6x + 9 - 9) - 29 \][/tex]
[tex]\[ f(x) = -4((x - 3)^2 - 9) - 29 \][/tex]
6. Distribute the \(-4\) through the parentheses:
[tex]\[ f(x) = -4(x - 3)^2 + 4 \cdot 9 - 29 \][/tex]
[tex]\[ f(x) = -4(x - 3)^2 + 36 - 29 \][/tex]
[tex]\[ f(x) = -4(x - 3)^2 + 7 \][/tex]
7. Now, we have the function in vertex form:
[tex]\[ f(x) = -4(x - 3)^2 + 7 \][/tex]
8. The vertex of this function \((h, k)\) gives the maximum (or minimum) height of the parabolic path. In this case, since the coefficient of \((x - h)^2\) is negative (-4), the parabola opens downward, indicating a maximum point.
The vertex is \((3, 7)\), so the maximum height of the water is given by the \(k\) value, which is \(7\) feet.
Therefore, the correct answer is:
[tex]\[ -4(x - 3)^2 + 7 \][/tex]
The maximum height of the water is [tex]\(7\)[/tex] feet.
