An archer releases an arrow from a shoulder height of 1.39 m. When
the arrow hits the target 18 m away, it hits point A. When the target is
removed, the arrow lands 45 m away. Find the maximum height of the
arrow along its parabolic path. please help!!I know my points are (0,1.39),(18, 1.684),(45,0) but I don't know how to put them in a quadratic equation.
let the equation be
y = ax^2 + bx + c
you have 3 points
(0,1.39) ---> 1.39 = 0 + 0 + c , ahh c = 1.39
(18,1.684) --> 1.684 = 324a + 18b + 1.39
324a + 18b = .258 --> 162a + 9b = .129 , #1
(45,0) ----->2025a + 45b + 1.39 = 0
2025a + 45b = -1.39
405a + 9b = -.278 , #2
#2 - #1 ---> 243a = -.407
a = -.001675
b = .0448
c = 1.39
y = -.001675x^2 + .0448x + 1.39 (rounded off a bit)
btw, Wolfram agrees with my answer
http://www.wolframalpha.com/input/?i=quadratic+through+%280%2C1.39%29%2C%2818%2C1.684%29%2C+%2845%2C0%29
comment: I don't know where you got those points from, but all 3 satisfy the equation I found
I was expecting "a" to be -4.9 according to gravity
We know the trajectory is along the parabola
y = 1.39 + ax + bx^2
and that
y(18) = 1.684
y(45) = 0
So,
1.39 + 18a + 324b = 1.684
1.39 + 45a + 2025b = 0
a = 0.0478
b = -.00175
y = -.00175x^2 + .0478x + 1.39
max altitude is reached at (13.657,1.716)
Thank u...But how did u get the 162a+9b=.129??
To find the maximum height of the arrow along its parabolic path, we can use the equation of a parabola in the form of y = ax^2 + bx + c. We need to find the coefficients a, b, and c by using the three given points: (0, 1.39), (18, 1.684), and (45, 0).
Step 1: Substitute the coordinates of the points into the equation y = ax^2 + bx + c one by one to get a system of three equations.
For the point (0, 1.39):
1.39 = a(0)^2 + b(0) + c
1.39 = c
So we have the first equation: c = 1.39.
Step 2: Substitute the coordinates of the second point (18, 1.684) into the equation y = ax^2 + bx + c.
1.684 = a(18)^2 + b(18) + 1.39 [using the value of c from the previous step]
Rearrange the equation to isolate the remaining variables:
a(18)^2 + b(18) = 1.684 - 1.39
(18^2)a + 18b = 0.294
So we have the second equation: (18^2)a + 18b = 0.294.
Step 3: Substitute the coordinates of the third point (45, 0) into the equation y = ax^2 + bx + c.
0 = a(45)^2 + b(45) + 1.39 [using the value of c from Step 1]
Rearrange the equation:
(45^2)a + 45b + 1.39 = 0
So we have the third equation: (45^2)a + 45b + 1.39 = 0.
We now have a system of three equations with three variables (a, b, c):
c = 1.39
(18^2)a + 18b = 0.294
(45^2)a + 45b + 1.39 = 0
Solving this system of equations will give us the values of a, b, and c, which can then be used to determine the maximum height of the arrow.
divided through by 2, just to get 9b for eliminating with #2.
algebra I, babe.