A stunt motorcyclist makes a jump from one ramp 20 feet off the ground. The jump between ramps can be modeled by y= -1/640x^2 + 1/4x +20 where x is the horizontal distance (in feet) and y is the height above ground (in feet).
I think a is 20 feet, but I don't know how to do b, c, or d. Please show me how to do the problems.
A. What is the motorcyclists height when he lands on the ramp?
B. What is the distance d between the ramps?
C. What is the horizontal distance h the motorcyclist has traveled when it reaches its maximum height?
D. What is the motorcyclists maximum height k above the ground
done here
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To solve these problems, we need to work with the given equation y = -1/640x^2 + 1/4x + 20. Here's how you can find the answers to each question:
A. To determine the height when the motorcyclist lands on the ramp, we need to find the value of y when x is equal to 0 (as the motorcyclist lands on the ramp, the horizontal distance is zero). Substitute x = 0 into the equation:
y = -1/640(0)^2 + 1/4(0) + 20
y = 0 + 0 + 20
y = 20
Therefore, the height when the motorcyclist lands on the ramp is 20 feet.
B. The distance d between the ramps can be found by finding the x-values when y = 0 (as the height above the ground is zero, indicating the motorcyclist is at ground level between the ramps). Set the equation to 0 and solve for x:
-1/640x^2 + 1/4x + 20 = 0
To solve this quadratic equation, you can either factor it or use the quadratic formula. For simplicity, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values a = -1/640, b = 1/4, and c = 20:
x = (-(1/4) ± √((1/4)^2 - 4(-1/640)(20))) / 2(-1/640)
Simplifying this expression will give you the two values of x. Take the positive value (since distance cannot be negative) to find the distance between the ramps, d.
C. To find the horizontal distance h when the motorcyclist reaches their maximum height, we need to find the x-value that corresponds to the vertex of the parabolic equation. The equation is in the form y = ax^2 + bx + c, where a = -1/640, b = 1/4, and c = 20.
The x-coordinate of the vertex is given by x = -b/2a. Substituting the values, we get:
x = - (1/4) / (2*(-1/640))
Simplifying this expression will give you the horizontal distance h.
D. Lastly, to find the maximum height k above the ground, we need to find the y-value of the vertex. Substitute the x-coordinate from part C into the equation:
y = -1/640( x-coordinate )^2 + 1/4( x-coordinate ) + 20
This will give you the maximum height k above the ground.
By following these steps, you should be able to solve all of the given problems related to the stunt motorcyclist's jump.