Final answer:
The maximum height reached by the ball is 76 feet at t=2 seconds. The ball takes 4 seconds to hit the ground after it has been tossed upwards.
Step-by-step explanation:
The function h(t) = -16t^2 + 64t + 12 represents the height above the ground, in feet, of a ball t seconds after being tossed upwards from the rooftop of a house. To find the maximum height reached by the ball, we need to complete the square or use the vertex form of a quadratic equation to determine the vertex of the parabola, since the coefficient of t^2 is negative, indicating that the parabola opens downwards and the vertex will give us the maximum height. By completing the square or using the vertex formula, we find that the maximum height is reached at t = 2 seconds. Plugging this value back into the height function, we get the maximum height as h(2) = -16(2)^2 + 64(2) + 12, which simplifies to 76 feet. To find the time it takes for the ball to hit the ground, we need to solve for t when h(t) equals zero. We end up using the quadratic formula given the quadratic nature of the equation, and we disregard the negative time to focus on the positive root which is the realistic scenario for our problem. Since the positive root of the quadratic equation is t = 4 seconds, we determine the ball takes 4 seconds to hit the ground after being tossed.