Final answer:
To find the time it takes the softball to reach the catcher's mitt, we solve the equation h(t) = -4.9t² + 19.6t for when the height is zero. Factoring out t gives two solutions, t = 0 (when the ball is thrown) and t = 4 seconds, which is the positive value indicating the time it will take for the ball to reach the ground.
Step-by-step explanation:
To determine how many seconds it will take the softball to reach the catcher's mitt on the ground at home plate, we need to solve the given quadratic equation for time t when the height h(t) is equal to zero. The equation provided is h(t) = -4.9t² + 19.6t. This equation models the motion of the softball as it follows a parabolic path under the influence of gravity (ignoring air resistance). The ball will reach the ground when its height is zero, hence we need to find the positive value of t that satisfies the equation -4.9t² + 19.6t = 0.
To solve the equation we can factor out t, giving us:
t(-4.9t + 19.6) = 0.
This gives us two solutions, t = 0 and -4.9t + 19.6 = 0. The first solution, t = 0, corresponds to the moment the ball is thrown. The second solution is found by rearranging the equation to t = 19.6 / 4.9, which simplifies to t = 4 seconds. This is the positive value of t that we are interested in, since time cannot be negative. Therefore, it will take the softball 4 seconds to reach the catcher's mitt.