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?
Assuming this is a calculus question and not a graphical one, you need to solve a simple differential equation in d:
h = 2d - (d^2)/5, so
dh/dd = 2 - 2d/5 = 0 at the maximum, so d=5. But at d=5, h= 2*5 - (5^2)/5 = 10 - 5 = 5. So the maximum height is 5 feet. Not very high, is it? Do check it and see if you agree.
To find the maximum height the dolphin can jump, we need to determine the vertex of the parabolic equation that models the dolphin's path. The vertex of a parabola is also the highest or lowest point of the curve.
The equation for the height of the dolphin can be represented as h = -0.2d^2 + 2d. Here, h represents the height of the dolphin in feet, and d represents the horizontal distance.
To find the vertex, we can use the formula: d = -b / 2a, where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c.
In our case, 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 d at the vertex is 5 feet.
Now, to find the maximum height, we substitute this value of d back into the equation for h:
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.