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.
Honestly god bless these people who ask these exact math questions almost a decade before you do. Really saving me right now.
D is 30 feet above the ground.
correct, that is what I had
30
A) If the two ramps have the same height, then your are correct at 20 ft. The height as he leaves the ramp is 20 , so hopefully he lands on the 2nd ramp at a height of 20 ft
B) set y = 20
- 1/640x^2 + (1/4)x + 20 = 20
multiply by -640
x^2 - 160x = 0
x(x - 160) = 0
x = 0 or x = 160
so the distance from take-off to landing is from 0 to 160.
So the distance between them is 160 ft
C) the max height would be at the vertex , that is, when the x = 80
so he reaches the highest point at the 80 ft mark
d) to get the max height, plug in x = 80 into the original equation
I will let you do that, let me know what you get.
A. To find the height when the motorcyclist lands on the ramp, we need to find the value of y when x is equal to the distance between the ramps. Let's call this distance d. So, substitute x = d into the equation y = -1/640x^2 + 1/4x + 20:
y = -1/640(d^2) + 1/4(d) + 20
This equation will give you the height when the motorcyclist lands on the ramp.
B. To find the distance d between the ramps, we need to determine where the height is equal to zero. Set the equation equal to zero and solve:
0 = -1/640x^2 + 1/4x + 20
This equation will give you the horizontal distance between the ramps.
C. To find the horizontal distance h when the motorcyclist reaches its maximum height, we need to find the x-coordinate of the vertex of the parabola. The equation for the x-coordinate of the vertex of a parabola in the form y = ax^2 + bx + c is given by:
x = -b / (2a)
In this case, a = -1/640 and b = 1/4. Plug these values into the equation to find the horizontal distance.
D. To find the maximum height k, we need to find the value of y when x is equal to the x-coordinate of the vertex. Substitute the x-coordinate into the equation to find the maximum height above the ground.
To find the answers to the given questions, we will need to use the equation provided: y = -1/640x^2 + 1/4x + 20.
A. What is the motorcyclist's height when he lands on the ramp?
When the motorcyclist lands on the ramp, the height above the ground will be zero (y = 0). To find the x-coordinate (distance) when y = 0, we set the equation equal to zero and solve for x:
0 = -1/640x^2 + 1/4x + 20
We can then solve this equation using algebraic methods like factoring or the quadratic formula to find the value(s) of x. Once we have the value(s) of x, we can substitute them back into the equation to determine the corresponding height (y).
B. What is the distance d between the ramps?
The distance between the ramps can be found by determining the horizontal distance (x) when the height (y) is zero. We already have this equation:
0 = -1/640x^2 + 1/4x + 20
Similar to Part A, we need to solve this equation to find the value(s) of x. The value(s) of x will give us the distance between the ramps.
C. What is the horizontal distance h the motorcyclist has traveled when it reaches its maximum height?
To find the horizontal distance (h) when the motorcyclist reaches their maximum height, we need to determine the vertex of the parabolic equation. The vertex of a parabolic equation in the form y = ax^2 + bx + c is given by the formula:
x = -b / (2a)
In our equation, y = -1/640x^2 + 1/4x + 20, a = -1/640, and b = 1/4. We can substitute these values into the formula to find x. Once we have the value of x, we can calculate the horizontal distance (h) traveled.
D. What is the motorcyclist's maximum height (k) above the ground?
To find the maximum height (k) above the ground, we need to substitute the value of x from Part C into the equation y = -1/640x^2 + 1/4x + 20. This will give us the corresponding y-coordinate for the maximum height.
By following these steps, you should be able to determine the answers to the questions. Make sure to solve the equations algebraically or using the quadratic formula to obtain the values of x where necessary.