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
To find the maximum height, we need to find the vertex of the equation h(t) = -16t^2 + 24t + 50.
The vertex of a quadratic equation in the form ax^2 + bx + c is given by the formula: x = -b / (2a).
In this case, a = -16 and b = 24. Plugging these values into the formula, we get:
t = -24 / (2 * -16)
t = -24 / -32
t = 0.75
Therefore, the ball will reach its maximum height at t = 0.75 seconds.
To find the maximum height h(t), substitute t = 0.75 into the equation:
h(0.75) = -16(0.75)^2 + 24(0.75) + 50
Calculating this expression, we get:
h(0.75) = -9 + 18 + 50
h(0.75) = 59
Therefore, the ball will reach a maximum height of 59 feet.
To determine how far above the ground the ball will be, we need to find the value of h(t) when t = 0 (the initial height when the ball is thrown).
Substituting t = 0 into the equation:
h(0) = -16(0)^2 + 24(0) + 50
h(0) = 50
Therefore, the ball will be 50 feet above the ground when it is thrown.
To find the time when the ball reaches its maximum height, we need to find the vertex of the quadratic equation h(t) = -16t^2 + 24t + 50. The vertex of a parabola represents the maximum or minimum point of the curve.
The equation for the vertex of a quadratic function is given by t = -b / (2a), where a, b, and c are the coefficients in the quadratic equation h(t) = at^2 + bt + c.
In this case, a = -16 and b = 24. Plugging these values into the formula, we can find the time:
t = -24 / (2 * -16)
t = -24 / -32
t = 0.75 seconds
Therefore, the ball reaches its maximum height after 0.75 seconds.
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) = -9 + 18 + 50
h(0.75) = 59 feet
So, the ball will be 59 feet above the ground when it reaches its maximum height.
a. V = Vo + g*t = 0.
24 + (-32)t = 0,
t = 0.75 s.
b. Use your given Eq to find ht.