Can you please help me with a math problem
Suppose the ball was 3.24 feet above ground when it was hit, and that it reached a maximum height of approximately 102.51 feet when it had traveled a ground distance of approximately 214.26 feet. The ball lands after traveling a ground distance of approximately 432 feet.
Find an equation of the form y = C(x-z1)(x-z2) where z1 and z2 are the zeros (or roots) of the quadratic polynomial (or x-intercepts of the graph) and C is a scaling constant that needs to be determined.
Thanks!
the vertex is at (214.26,102.51)
So, y = C(x-214.26)^2 + k
102.51 = C(0)^2 + k
so, k = 102.51
y = C(x-214.26)^2 + 102.51
0 = C(432-214.26)^2 + 102.51
C = -0.002162
y = -0.002162(x-214.26)^2 + 102.51
y = -.002162x^2 + .92646x + 3.25831
y = -.002162(x+3.4885)(x-432)
Why are these poor models for a parabola, Where a ball starts a a certain point and then is hit to reach a maximum height (vertex) and then lands at a certain point
i. y = -0.002x(x - 437.1)
ii. y = -0.5x + 216x + 3
iii. y = -0.002x + 0.879x + 3.981
iv. y = -0.002x + 0.8732x - 3.981
Thanks!
To find an equation in the form y = C(x-z1)(x-z2), we need to find the values of z1 and z2. These values represent the x-intercepts or roots of the quadratic polynomial.
Let's start by establishing some key information from the given problem:
1. The ball was 3.24 feet above the ground when it was hit.
2. The ball reached a maximum height of approximately 102.51 feet when it had traveled a ground distance of approximately 214.26 feet.
3. The ball lands after traveling a ground distance of approximately 432 feet.
Based on this information, we can determine the equation of the quadratic polynomial.
STEP 1: Vertex Form
The vertex form of a quadratic equation is given by y = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex. We can use this form because we have the vertex's y-coordinate.
Using the given values, the vertex form equation becomes:
y = a(x - 214.26/2)^2 + 102.51
Simplifying, we get:
y = a(x - 107.13)^2 + 102.51
STEP 2: Finding the Scaling Constant, C
Since we want the equation in factored form (y = C(x-z1)(x-z2)), we need to multiply the quadratic equation by a constant, C.
Using the fact that the ball was 3.24 feet above the ground when it was hit, we can substitute the value for y and solve for C.
3.24 = a(0 - 107.13)^2 + 102.51
Simplifying, we get:
3.24 = a(107.13)^2 + 102.51
Now, we can solve for 'a' by isolating it:
a(107.13)^2 = 3.24 - 102.51
a(107.13)^2 = -99.27
a = -99.27 / (107.13)^2
STEP 3: Determining the X-Intercepts
To find the x-intercepts or zeros of the quadratic polynomial, we need to determine when y becomes zero. Since the ball lands after traveling a ground distance of approximately 432 feet, we can substitute y = 0 and x = 432 into the equation:
0 = C(432 - z1)(432 - z2)
0 = C(432 - z1)(432 - z2)
Now, we have the equation in factored form, which is what we desired. The values of z1 and z2 are the x-intercepts or roots of the quadratic polynomial.