Question

1. path of a diver

the path of a diver is modeled by (f(x)=-\frac{4}{9}x^{2}+\frac{24}{9}x + 12) where (f(x)) is the height (in feet) and (x) is the horizontal distance (in feet) from the end of the diving - board. what is the maximum height of the diver?

Explanation:

Step1: Identify the function type

The function $f(x)=-\frac{4}{9}x^{2}+\frac{24}{9}x + 12$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a=-\frac{4}{9}$, $b = \frac{24}{9}$, and $c = 12$.

Step2: Find the x - coordinate of the vertex

The x - coordinate of the vertex of a quadratic function $y=ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. Substitute $a =-\frac{4}{9}$ and $b=\frac{24}{9}$ into the formula:
\[x=-\frac{\frac{24}{9}}{2\times(-\frac{4}{9})}=-\frac{\frac{24}{9}}{-\frac{8}{9}}=\frac{24}{9}\times\frac{9}{8}=3\]

Step3: Find the maximum value of the function

Substitute $x = 3$ into the function $f(x)=-\frac{4}{9}x^{2}+\frac{24}{9}x + 12$:
\[f(3)=-\frac{4}{9}\times3^{2}+\frac{24}{9}\times3+12\]
\[=-\frac{4}{9}\times9+\frac{24}{9}\times3 + 12\]
\[=- 4+8 + 12\]
\[=16\]

Answer:

16 feet