The path of a diver is approximated by the following, where y is the height (in feet) and x is the horizontal distance (in feet) from the end of the diving board.
y= (-4/9)x^2+(24/9)x+13
What is the maximum height of the diver in feet?
as with all ax^2+bx+c, the vertex is at (-b/2a, c - b^2/4a)
To find the maximum height of the diver, we need to determine the vertex of the quadratic equation represented by the given function.
The vertex of a quadratic function in the form of y = ax^2 + bx + c can be found using the formula:
x = -b / (2a)
Comparing the given equation to the standard form, we see that a = -4/9 and b = 24/9.
Now, let's substitute these values into the formula:
x = - (24/9) / (2 * (-4/9))
Simplifying further:
x = - (8/3) / (-8/9)
x = 3/9
x = 1/3
To find the maximum height, substitute this value of x back into the equation:
y = (-4/9)(1/3)^2 + (24/9)(1/3) + 13
y = (-4/9)(1/9) + (24/9)(1/3) + 13
y = (-4/81) + (8/9) + 13
y = (32/81) + (8/9) + 13
y = (32/81) + (8/9) + 117/9
y = (32/81) + (72/81) + 117/9
y = (104/81) + 117/9
y = (104/81) + (1053/81)
y = (1157/81)
Therefore, the maximum height of the diver is approximately 14.27 feet.
To find the maximum height of the diver, we need to determine the vertex of the given quadratic equation. The vertex of a quadratic equation in the form y = ax^2 + bx + c can be found using the formula x = -b/(2a).
In the given equation, y = (-4/9)x^2 + (24/9)x + 13, we can see that a = -4/9 and b = 24/9.
Using the formula x = -b/(2a), we can calculate the x-coordinate of the vertex:
x = -[(24/9)] / [2 * (-4/9)]
x = -24/18
x = -4/3
Now, substitute this value of x into the equation to find the y-coordinate of the vertex:
y = (-4/9) * [(-4/3)]^2 + (24/9) * (-4/3) + 13
y = (-4/9) * (16/9) - (32/9) + 13
y = -64/81 - 32/9 + 13
y = -64/81 - 288/81 + 1053/81
y = 701/81
Therefore, the maximum height of the diver is 701/81 feet.