A soccer ball is kicked upward from the ground and its height (in metres), t seconds after it is kicked, is given by the equation h(t)= -49t^2 + 19t. What wiil be the maximum height attained by the ball? And how much does it take for the ball to hit the ground
To find the maximum height attained by the ball, we need to determine the vertex of the parabolic equation h(t) = -49t^2 + 19t. The vertex of a parabola is the highest or lowest point on its graph.
First, let's find the time it takes for the ball to hit the ground. We know that the ball hits the ground when its height (h) is zero.
Setting h(t) = 0, we can solve the equation -49t^2 + 19t = 0.
Factoring out t, we get t(-49t + 19) = 0.
This equation is satisfied when either t = 0 (initial time) or -49t + 19 = 0.
Solving -49t + 19 = 0, we get t = 19/49.
So, it takes the ball 19/49 seconds to hit the ground.
Now, let's find the maximum height. The vertex of a parabola can be found using the formula t = -b / (2a), where a and b are coefficients of the quadratic equation.
In this case, a = -49 and b = 19.
Using the formula t = -b / (2a), we get t = -19 / (2 * -49).
Simplifying further, t = 19 / 98.
Substituting this value of t into the original equation h(t), we get
h(19/98) = -49 * (19/98)^2 + 19 * (19/98).
Calculating this expression yields approximately h(max) = 9.39 meters.
Therefore, the maximum height attained by the ball is approximately 9.39 meters, and it takes approximately 19/49 seconds for the ball to hit the ground.