Question

Stewart, Oswaldo, Kevin, and Flynn go to a soccer day at the FC Dallas’ arena, Toyota Stadium, in

Frisco, Texas. The coach has a computer and video system that can track the height and distance of
their kicks. All four soccer players are practicing up-field kicks, away from the goal.
Stewart goes first and takes a kick starting 12 yards out from the goal. His kick reaches a maximum
height of 17 yards and lands 48 yards from the goal.
Oswaldo goes next and the computer gives the equation of the path of his kick as y = -x 2 + 14x - 24,
where y is the height of the ball in yards and x is the horizontal distance of the ball from the goal line in
yards.
After Kevin takes his kick, the coach gives him a printout of the path of the ball.

Finally, Flynn takes his kick but the computer has a problem and can only give him a partial table of
data points of the ball’s trajectory.
Flynn’s Table:
Distance from the
goal line in yards

10 11 12 13 14 15 16 17 18 19 20

Height in yards 0 4.7 8.75 12.2 15 17.2 18.75 19.7 20 19.7 18.75
The computer is still not working but Stewart, Oswaldo, Kevin, and Flynn want to know who made the
best kick.

Answer 1

To compare the kicks, we need to determine which kick goes the farthest and/or highest. Let's analyze each kick one by one.

Stewart's kick:

Starting 12 yards out from the goal, Stewart's kick reaches a maximum height of 17 yards and lands 48 yards from the goal. We can assume that the ball lands at the same height as it was kicked. Therefore, the horizontal distance traveled by the ball is:

d = 48 - 12 = 36 yards

The total distance traveled by the ball is the hypotenuse of a right triangle with legs of 36 and 17 yards. Using the Pythagorean theorem, we can find the distance traveled by Stewart's kick:

distance = sqrt(36^2 + 17^2) ≈ 40.48 yards

Oswaldo's kick:

According to the equation given, the height of the ball (in yards) at any horizontal distance (in yards) from the goal is given by:

y = -x^2 + 14x - 24

We want to find the maximum height reached by the ball, so we need to find the vertex of the parabolic path. The x-coordinate of the vertex is given by:

x = -b / 2a = -14 / (-2) = 7

The maximum height is the y-coordinate of the vertex:

y = -(7)^2 + 14(7) - 24 = 25

Therefore, Oswaldo's kick reaches a maximum height of 25 yards.

To find the horizontal distance traveled by the ball, we need to find the x-intercepts of the parabolic path. Setting y = 0, we get:

0 = -x^2 + 14x - 24

Solving for x using the quadratic formula, we get:

x = (14 ± sqrt(14^2 - 4(-1)(-24))) / (2(-1)) ≈ 2.63, 11.37

Therefore, the ball lands at a horizontal distance of approximately 2.63 or 11.37 yards from the goal. The total distance traveled by the ball is the sum of the horizontal distance and the maximum height:

distance = 11.37 + 25 ≈ 36.37 yards

Kevin's kick:

We don't have the equation or data for Kevin's kick, so we can't determine the maximum height or the total distance traveled by the ball.

Flynn's kick:

We have a partial table of data points for Flynn's kick. We can plot the points on a graph and draw a curve that fits the data points. Here is the graph:

Flynn's Kick

The curve appears to be a parabolic path, so we can assume that the equation for the path is:

y = ax^2 + bx + c

To find the coefficients a, b, and c, we need to solve a system of equations using three data points. Let's use the data points (12, 8.75), (15, 17.2), and (18, 19.7).

Using the first data point, we get:

8.75 = a(12)^2 + b(12) + c

Using the second data point, we get:

17.2 = a(15)^2 + b(15) + c

Using the third data point, we get:

19.7 = a(18)^2 + b(18) + c

Solving this system of equations using a calculator or matrix methods, we get:

a ≈ 0.0571

b ≈ -1.

Hope this helped, I'm sorry if it didn't. If you need more help, ask me! :]