A ball is thrown from the top of a 50-ft building
with an upward velocity of 24 ft/s. When will it
reach its maximum height? How far above the
ground will it be? Use the equation
h(t)= -16t^2 + 24t + 50
as with any parabola "at^2+bt+c" the vertex is at
(-b/2a, c - b^2/4a)
To find the time it takes for the ball to reach its maximum height, we need to find the vertex of the quadratic equation h(t) = -16t^2 + 24t + 50.
The vertex of a quadratic equation in the form h(t) = at^2 + bt + c is given by the formula t = -b/2a.
In this case, a = -16 and b = 24, so plugging these values into the formula, we have:
t = -24 / (2 * -16)
t = -24 / -32
t = 0.75 seconds
Therefore, it will take 0.75 seconds for the ball to reach its maximum height.
To find the height at this time, we substitute t = 0.75 into the equation h(t):
h(0.75) = -16(0.75)^2 + 24(0.75) + 50
h(0.75) = -16(0.5625) + 18 + 50
h(0.75) = -9 + 18 + 50
h(0.75) = 59 feet
Therefore, the ball will reach a maximum height of 59 feet above the ground.
To find the time when the ball reaches its maximum height, we need to determine when the vertical velocity of the ball becomes zero. At the maximum height, the ball momentarily stops going up and starts to come back down.
The equation provided is a height-time equation: h(t) = -16t^2 + 24t + 50, where h(t) represents the height of the ball at time t.
To find the time when the velocity is zero, we need to find the vertex of the parabolic equation. The vertex of a parabola in the form of y = ax^2 + bx + c is given by the formula: t = -b / (2a).
In this case, a = -16 and b = 24. Substituting these values into the formula, we have t = -24 / (2 * -16). Simplifying this expression, we find t = 3 / 4.
Therefore, the ball reaches its maximum height after 3/4 of a second.
To determine how far above the ground the ball will be at its maximum height, we can plug this value of t back into the equation h(t) = -16t^2 + 24t + 50.
h(3/4) = -16(3/4)^2 + 24(3/4) + 50
= -16(9/16) + 72/4 + 50
= -9 + 18 + 50
= 59
Therefore, the ball will be 59 feet above the ground at its maximum height.