Question

Richard tosses a ball into the air. the function h(t) = −5^2 10t 6 gives the approximate height h, in meters, of the ball t seconds after he tosses it. does the ball reach a height of 12 m? after how many seconds will it reach the ground?

Answer 1

The question involves using a quadratic function to determine if a ball reaches a height of 12 meters and when it hits the ground. Assuming a typo in the function, we approximate that the positive root from the quadratic equation gives the time when the ball hits the ground, considering projectile motion principles.

Step-by-step explanation:

The question involves determining whether the ball reaches a certain height and finding the time at which the ball reaches the ground based on the motion of the ball described by the quadratic function h(t) = -5t^2 + 10t + 6. To establish whether the ball reaches a height of 12 meters, you would need to solve the equation -5t^2 + 10t + 6 = 12. However, the provided function seems to have a typo, so please provide the correct function for a more accurate answer.

To find when the ball reaches the ground, we set the function equal to zero and solve for t. Assuming the correct quadratic equation is h(t) = -5t^2 + 10t + 6, we solve -5t^2 + 10t + 6 = 0 with the quadratic formula, yielding two possible times, of which the positive value will be the time it takes for the ball to hit the ground. The quadratic formula is t = (-b ± √(b^2 - 4ac))/(2a), where a, b, and c are the coefficients from the quadratic equation at^2 + bt + c = 0.

Given that the calculations demonstrate that the ball reaches the height of 10 meters at two points in time based on the similar nature of projectile motion, we can use the same reasoning to determine if it reaches 12 meters. By solving the provided equation, we can find the two times it reaches this height, one on the way up and one on the way down.

When the ball hits the ground, its height is 0. We therefore set the equation to zero: -5t^2 + 10t + 6 = 0 and use the quadratic formula or factoring to solve for t. The relevant root is the positive one because time in this context cannot be negative. The larger value of t after calculating should be taken as the time it takes for the ball to hit the ground while the lower value corresponds to an earlier time in the ball's flight.