The path of a diver is modeled by f(x) = − 4 9 x2 + 24 9 x + 11 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?

We have been given that the path of a diver is modeled by [tex]f(x)=-\frac{4}{9}x^2+\frac{24}{9}x + 11[/tex], where f(x) is the height (in feet) and x is the horizontal distance (in feet) from the end of the diving board. We are asked to find the maximum height of the diver.

Since our given function is downward opening parabola, so its maximum point will be vertex.

To find maximum height of the diver, we need to figure out y-coordinate of vertex.

First of all, we will fund x-coordinate of vertex using formula [tex]-\frac{b}{2a}[/tex] as:

[tex]x=\frac{-b}{2a}=\frac{-\frac{24}{9}}{2*\frac{-4}{9}}=\frac{\frac{24}{9}}{\frac{8}{9}}=\frac{24*9}{8*9}=3[/tex]

Let us find y-coordinate of vertex by substituting [tex]x=3[/tex] in the given function as:

[tex]f(3)=-\frac{4}{9}\cdot (3)^2+\frac{24}{9}\cdot (3) + 11[/tex]

[tex]f(3)=-\frac{4}{9}\cdot 9+\frac{24}{3}\cdot (1) + 11[/tex]

[tex]f(3)=-4+8+ 11[/tex]

[tex]f(3)=15[/tex]

Therefore, the maximum height of the diver is 15 feet.