a bootlenose dolphin jumps out of the water. the path the dolphin travels can be modeled by h=-0.2d^2+2d, where h represents the height of the dolphin in feet and d represents the horizontal distance.
What is the maximum height the dolphin can jump?
h(d)=-0.2d^2+2d
To find the maximum (or minimum), we find the derivative with respect to d and equate to zero.
h'(d) = -0.4d+2 = 0
=>
d = 2/0.4 = 5 ft.
Check that this is the maximum:
method 1:
a parabola with a negative coefficient has only one maximum and no minimum.
method 2:
check sign of h"(d) where d=5'
h"(d) = -0.4 <0 => maximum.
To find the maximum height the dolphin can jump, we need to find the vertex of the quadratic equation h = -0.2d^2 + 2d.
The equation is in the form of h = ax^2 + bx + c, where a = -0.2, b = 2, and c = 0.
To find the vertex, we use the formula:
d = -b / (2a)
Plugging in the values:
d = -2 / (2 * (-0.2))
d = -2 / (-0.4)
d = 5
So, the dolphin will reach its maximum height at a horizontal distance of 5 feet.
To find the maximum height, substitute this value back into the equation:
h = -0.2(5)^2 + 2(5)
h = -0.2(25) + 10
h = -5 + 10
h = 5 feet
Therefore, the maximum height the dolphin can jump is 5 feet.
To find the maximum height the dolphin can jump, we need to determine the vertex of the quadratic equation h = -0.2d^2 + 2d.
The vertex of a quadratic equation in the form h = ax^2 + bx + c is given by the formula:
x = -b / (2a)
In our equation, a = -0.2 and b = 2. Plugging these values into the formula, we get:
d = -2 / (2*(-0.2))
d = -2 / (-0.4)
d = 5
So, the horizontal distance at the vertex is 5 feet.
Now, we can substitute this value of d back into the equation to find the maximum height:
h = -0.2(5)^2 + 2(5)
h = -0.2(25) + 10
h = -5 + 10
h = 5
Therefore, the maximum height the dolphin can jump is 5 feet.