Math Problem Statement

The path of a diver is modeled by f(x) = -(4/9)x^2 + (24/9)x + 9, 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 (in ft) of the diver?

Solution

The function modeling the diver's path is given as:

f(x)=49x2+249x+9f(x) = -\frac{4}{9}x^2 + \frac{24}{9}x + 9

This is a quadratic function of the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where a=49a = -\frac{4}{9}, b=249b = \frac{24}{9}, and c=9c = 9.

Since aa is negative, the parabola opens downwards, and the vertex of the parabola represents the maximum point. To find the maximum height, we need to determine the x-coordinate of the vertex using the formula:

x=b2ax = -\frac{b}{2a}

Substitute the values of aa and bb:

x=2492×49=24989=249×98=248=3x = -\frac{\frac{24}{9}}{2 \times -\frac{4}{9}} = \frac{\frac{24}{9}}{\frac{8}{9}} = \frac{24}{9} \times \frac{9}{8} = \frac{24}{8} = 3

So, the horizontal distance at which the maximum height occurs is x=3x = 3.

Now, substitute x=3x = 3 into the original equation to find the maximum height:

f(3)=49(3)2+249(3)+9f(3) = -\frac{4}{9}(3)^2 + \frac{24}{9}(3) + 9

Simplify:

f(3)=49(9)+249(3)+9=4+8+9=13f(3) = -\frac{4}{9}(9) + \frac{24}{9}(3) + 9 = -4 + 8 + 9 = 13

Thus, the maximum height of the diver is 13 feet.

Would you like any more details or have further questions on this? Here are some related questions to consider:

  1. What is the significance of the vertex in a quadratic function?
  2. How do you determine whether a parabola opens upwards or downwards?
  3. How does the discriminant of a quadratic equation relate to its graph?
  4. What would happen to the maximum height if the coefficients were changed?
  5. How do you find the axis of symmetry for a quadratic function?

Tip: The vertex formula x=b2ax = -\frac{b}{2a} works for any quadratic function and is a quick way to find the maximum or minimum point depending on the direction of the parabola!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Vertex of a Parabola

Formulas

f(x) = ax^2 + bx + c
Vertex formula: x = -b/2a

Theorems

Maximum height occurs at the vertex of a parabola

Suitable Grade Level

Grades 9-11