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