The path of a cliff diver as he dives into a lake is given by the equation
y= -(x-10)^2+75, where y meters is the divers height above the water and x meters is the horizontal distance travelled by the diver. What is the maximum hight the diver is above water?
How can I find this answer?
y = -(x - 10)^2 + 75.
y = a(x - h)^2 + k.
The above Eqs represent a parabola in
Vertex Form.
Since "a" is neg., the parabola opens downward and the vertex is the maximum
point on the curve.
The coordinates of the vertex are:
V(h , k) = V(10 , 75).
Max ht = k = 75 meters.
To find the maximum height the diver is above water, we need to determine the vertex of the equation. The vertex represents the maximum point or the highest point of the parabola.
The equation of the parabola is given as y = -(x - 10)^2 + 75. By comparing this equation with the standard form of a parabola, y = a(x - h)^2 + k, we can identify that the vertex is at the point (h, k). The value of h represents the horizontal shift or the x-coordinate of the vertex.
In our equation, the value of h is 10. Therefore, the vertex of the parabola is at the point (10, k). To find the value of k, we substitute 10 back into the equation:
y = -(10 - 10)^2 + 75
= -(0)^2 + 75
= -0 + 75
= 75
Hence, the maximum height the diver is above the water is 75 meters.
To find the maximum height the diver reaches above the water, you need to determine the vertex of the parabolic equation. The vertex represents the highest point of the parabola, which in this case represents the maximum height of the diver above water.
The equation of the parabola is y = -(x - 10)^2 + 75. This equation is written in vertex form, which is y = a(x - h)^2 + k. In this form, (h, k) represents the coordinates of the vertex.
Comparing the given equation to the vertex form, we can determine that the parabola is translated horizontally by 10 units to the right (h = 10) and translated vertically by 75 units upwards (k = 75).
Therefore, the coordinates of the vertex (h, k) are (10, 75). The x-coordinate, 10, represents the horizontal distance travelled by the diver, and the y-coordinate, 75, represents the maximum height the diver reaches above the water.
Hence, the maximum height the diver is above the water is 75 meters.