Question

When a baseball is thrown upward, its height h, in feet, is the function of time t, in seconds. If this function is described by the formula h(t)=-10t² +24t+5, what is the maximum height of the baseball reaches?

Answer 1

The maximum height reached by the baseball is calculated by finding the vertex of the quadratic equation representing the ball's motion. The vertex occurs at time t = 1.2 seconds, and the maximum height calculated by substituting this value into the equation is 19.4 feet.

Step-by-step explanation:

To determine the maximum height the baseball reaches when thrown upward, we need to analyze the quadratic function h(t) = -10t² + 24t + 5, which gives the height h of the baseball at any given time t. This is a parabolic motion described by a quadratic formula, where the coefficient of the t² term is negative, indicating that the parabola opens downwards. Therefore, the vertex of the parabola corresponds to the maximum height of the baseball.

The equation of the parabola is y = ax² + bx + c, and the maximum height is given at the vertex, which can be found using the formula t = -b/(2a). For our function, a is -10, and b is 24, so the time at which the maximum height is reached is t = -24/(2 × -10) = 1.2 seconds.

To find the maximum height, we substitute t = 1.2 seconds into the original equation: h(1.2) = -10(1.2)² + 24(1.2) + 5. After calculating, we find that the maximum height is h(1.2) = -10(1.44) + 28.8 + 5 = -14.4 + 28.8 + 5 = 19.4 feet. Thus, the maximum height reached by the baseball is 19.4 feet.