SOLUTION: A cannon fires a cannonball whose path is a parabola with vertex at the highest point of the path. If the cannonball lands 1600 feet from the cannon and the highest point reac

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: A cannon fires a cannonball whose path is a parabola with vertex at the highest point of the path. If the cannonball lands 1600 feet from the cannon and the highest point reac      Log On


   

Question 688908: A cannon fires a cannonball whose path is a parabola with vertex
at the highest point of the path. If the cannonball lands 1600 feet from the
cannon and the highest point reached is 3200 feet above the ground, find an
equation for the path of the cannonball.
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
You can relate 2 variables- +h+ plotted on
the vertical axis, and +d+ plotted on the horizontal
axis. I know that the 2 roots of the equation are:
+d+=+0+
+d+=+1600+
This is because +d+=+0+ is where the cannonball is
fired from and +d+=+3200+ is where it lands.
----------
The general form for a parabola is
+h+=+a%2Ad%5E2+%2B+b%2Ad+%2B+c+
You are given 2 points, ( 0,0 ) and ( 1600, 0 )
(0,0)
+0+=+a%2A0%5E2+%2B+b%2A0+%2B+c+
+0+=+c+
So now the equation looks like
+h+=+a%2Ad%5E2+%2B+b%2Ad+
-------------------
(1600,0)
+0+=+a%2A%281600%29%5E2+%2B+b%2A1600+
+a%2A%281600%29%5E2+=+-b%2A1600+
+1600a+=+-b+
+b+=+-1600a+
----------------
The vertex is at +-b%2F%282a%29%2C+h%5Bmax%5D+
+-b%2F%282a%29+=+1600a%2F%282a%29+
+-b%2F%282a%29+=+800+
You are given that +h%5Bmax%5D+=+3200+
+3200+=+a%2A800%5E2+%2B+b%2A800+
+4+=+800a+%2B+b+
Since +b+=+-1600a+
+4+=+800a+-+1600a+
+4+=+-800a+
+a+=+-1%2F200+
and
+b+=+-1600a+
+b+=+-1600%2A%28-1%2F200+%29+
+b+=+8+
The equation is:
+h+=+-%281%2F200%29%2Ad%5E2+%2B+8d+
------------------------
check answer:
Does it go through (0,0)?
+h+=+-%281%2F200%29%2Ad%5E2+%2B+8d+
+0+=+-%281%2F200%29%2A0%5E2+%2B+8%2A0+
+0+-+0+
OK
Does it go through (1600,0)?
+h+=+-%281%2F200%29%2Ad%5E2+%2B+8d+
+0+=+-%281%2F200%29%2A1600%5E2+%2B+8%2A1600+
+0+=+-2560000%2F200+%2B+12800+
++0+=+-12800+%2B12800+
+0+=+0+
OK
Is the vertex at ( 800, 3200 )?
+h+=+-%281%2F200%29%2Ad%5E2+%2B+8d+
+3200+=+-%281%2F200%29%2A800%5E2+%2B+8%2A800+
+4+=+-%281%2F200%29%2A800+%2B+8+
+%281%2F200%29%2A800+=+4+
+4+=+4+
OK
Here's a plot of the parabola.
+graph%28+600%2C+600%2C+-500%2C+1800%2C+-500%2C+3500%2C+-%281%2F200%29%2Ax%5E2+%2B+8x+%29+