The path of a diver is given by
y=-4/9 x^2 + 24/9x +12
where y is the height (in feet) and x is the horizontal distance from the end of the diving board (in feet). What is the maximum height of the diver?
max height occurs when x = -b/2a
Here, that means max height is where
x = -(24/9) / (-8/9) = 3
so, find y(3)
To find the maximum height of the diver, you need to determine the vertex of the parabolic equation given by y = -4/9 x^2 + 24/9x + 12. The vertex of a parabola is the point where it reaches its highest or lowest point.
The equation of a parabola is in the form y = ax^2 + bx + c. To find the vertex, you can use the formula:
x = -b / (2a)
In this case, a = -4/9 and b = 24/9. Plugging in these values into the formula, you get:
x = -(24/9) / (2 * (-4/9))
Simplifying this expression, you get:
x = -(8/3)
Now, substitute this value of x back into the equation to find the corresponding value of y:
y = -4/9 * (-(8/3))^2 + 24/9 * (-(8/3)) + 12
Simplifying this expression, you get:
y = -256/27 + 64/3 + 12
Combining the terms, you have:
y = -256/27 + 192/27 + 12
y = -256/27 + 192/27 + 324/27
y = 260/27
Therefore, the maximum height of the diver is 260/27 feet or approximately 9.63 feet.
To find the maximum height of the diver, we need to determine the vertex of the equation. The vertex of a quadratic equation in the form y = ax^2 + bx + c is given by the coordinates (-b/2a, f(-b/2a)), where a, b, and c are the coefficients of the equation.
In the given equation, y = (-4/9)x^2 + (24/9)x + 12, the coefficient of x^2 is -4/9 and the coefficient of x is 24/9.
To find the x-coordinate of the vertex, we can use the formula x = -b/2a.
Substituting the coefficients into the formula, we have:
x = -(24/9) / (2 * (-4/9))
Simplifying, we get:
x = -24/18 / -8/18
x = 24/8
x = 3
Now, to find the y-coordinate of the vertex, we substitute the value of x = 3 into the equation:
y = (-4/9)(3)^2 + (24/9)(3) + 12
Simplifying, we get:
y = (-4/9)(9) + (24/9)(3) + 12
y = -4 + 8 + 12
y = 16
Therefore, the maximum height of the diver is 16 feet.