SOLUTION: from the center of the 20yd (60ft) line, a football player attempts to make a field goal by kicking the ball directly toward the goal posts, which are 90ft away. The goal-post cros

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: from the center of the 20yd (60ft) line, a football player attempts to make a field goal by kicking the ball directly toward the goal posts, which are 90ft away. The goal-post cros      Log On


   

Question 211525: from the center of the 20yd (60ft) line, a football player attempts to make a field goal by kicking the ball directly toward the goal posts, which are 90ft away. The goal-post crossbar is 10 ft above the ground. The ball reaches its highest altitude of 32 ft at a point 48 ft from where it was kicked.
a. make a sketch showing the path of the football. if the point from which the ball is kicked is the origin of a coordinate plane, find the equation of the parabolic path of the football.
b. will the kicker make the field goal?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

The football is kicked at point (0,0).
The highest point (vertex) is at (48,32)
The goalpost crossbar is at (90,10).

We begin by plotting the vertex, and drawing
a line for the goalpost.



The equation of a parabola is

y=a%28x-h%29%5E2%2Bk where the vertex is %22%28h%2Ck%29%22

We substitute %22%28h%2Ck%29%22=%22%2848%2C32%29%22

y=a%28x-48%29%5E2%2B32

We can find a because the parabola goes
through the origin %22%280%2C0%29%22

So we substitute %22%28x%2Cy%29%22=%22%280%2C0%29%22

y=a%28x-48%29%5E2%2B32
0=a%280-48%29%5E2%2B32
0=a%28-48%29%5E2%2B32
0=a%282304%29%2B32
0=2304a%2B32
-32=2304a
%28-32%2F2304%29=a
-1%2F72=a

So the equation of the parabola is

y=%28-1%2F72%29%28x-48%29%5E2%2B32

We draw the parabola:



Oh,oh.  From the picture it appears that the
football is going to drop too low to go over
the goal post bar.  But we can't just go by the 
picture.  Let's calculate the height of the 
ball when x=90 and find out for sure:

y=%28-1%2F72%29%28x-48%29%5E2%2B32
y=%28-1%2F72%29%2890-48%29%5E2%2B32
y=%28-1%2F72%29%2842%29%5E2%2B32
y=%28-1%2F72%29%281764%29%2B32
y=%28-1764%2F72%29%2B32
y=-24.5%2B32
y=7.5

No, the field goal will not be made, because
the ball will be too low, and will go under
the goal post, since 7.5 feet is less
than 10 feet.

Edwin