Question

To begin a football game, a kicker has to kick off. The football follows the path h(x)=−130x(x−61)
, where h(x)
is the height of the football and x is the horizontal distance it has traveled in yards. Solve the equation to determine how far the ball will have traveled when it hits the ground.

0=−130x(x−61)

Answer 1

To determine how far the ball will have traveled when it hits the ground, we need to solve the equation given by the height function set to zero, i.e.,

\[
0 = -130x(x - 61)
\]

We can factor it as follows:

\[
0 = -130x(x - 61)
\]

This product equals zero when either of the factors is zero. Therefore, we can set each factor equal to zero:

1. \( -130 = 0 \) (This does not provide a solution since -130 is not equal to zero)
2. \( x = 0 \) (This solution represents the starting point of the kick)
3. \( x - 61 = 0 \)

From the third factor, we can solve for \( x \):

\[
x - 61 = 0 \\
x = 61
\]

Thus, the solutions to the equation are \( x = 0 \) and \( x = 61 \).

Since \( x=0 \) is the starting point of the kick, the ball will hit the ground when it has traveled \( x = 61 \) yards. Therefore, the ball travels 61 yards when it hits the ground.