Question

A boy throws a stone into the air from a cliff, and the height (h in meters) of the stone at any time (t in seconds) can be modeled by the equation h = -4.8t^2 + 16t + 45. How can we determine the time it takes for the stone to reach its maximum height, and what is the maximum height it reaches?

Answer 1

The time it takes for the stone to reach its maximum height can be calculated using the formula t = -b/(2a), which gives approximately 1.67 seconds. The maximum height reached by the stone can then be found by substituting this time back into the original quadratic equation, resulting in a maximum height of 53.3 meters.

Step-by-step explanation:

To determine the time it takes for the stone to reach its maximum height using the equation h = -4.8t² + 16t + 45, we need to find the vertex of the parabola represented by this quadratic equation. The time at which the maximum height is reached can be found using the formula t = -b/(2a) where a is the coefficient of t² (-4.8) and b is the coefficient of t (16). In this equation, the maximum height is the value of h at this time.

The t coordinate of the vertex (maximum height point) is found by plugging in the values into the formula: t = -16/(2 * -4.8) = 16/9.6 = 1.6667 seconds. To find the maximum height, we substitute this value back into the original equation, yielding h at maximum height: h = -4.8 * (1.6667)² + 16 * 1.6667 + 45.

Solving this, the maximum height the stone reaches is 53.3 meters.

Answer 2

The time it takes for the stone to reach its maximum height is 1.67 seconds and the maximum height it reaches is 50.37 meters.

Step-by-step explanation:

To determine the time it takes for the stone to reach its maximum height, we can use the vertex formula for a quadratic equation. The vertex formula is given by: t = -b / (2a), where a = -4.8 and b = 16. Substituting these values into the formula, we get t = -16 / (2 × -4.8) = 1.67 seconds.

To find the maximum height the stone reaches, we can substitute the value of t = 1.67 seconds into the equation h = -4.8t^2 + 16t + 45. Evaluating the equation, we get h = -4.8(1.67^2) + 16(1.67) + 45 = 50.37 meters.