An object in launched directly upward at 64 feet per second (ft/s) from a platform 80 feet high. This is modeled by the equation h = −16𝑡2 + 64𝑡 + 80.
To find the time it takes for the object to reach its maximum height, we need to determine the vertex of the given quadratic equation.
The equation for the object's height as a function of time is given as h = -16𝑡² + 64𝑡 + 80, where h represents the height and t represents the time.
The equation for the vertex of a quadratic function in the form ax² + bx + c is given as t = -b / (2a).
Comparing the given equation h = -16𝑡² + 64𝑡 + 80 with the standard form ax² + bx + c, we have a = -16 and b = 64.
Substituting the values into the vertex equation:
t = -b / (2a)
t = -64 / (2*(-16))
t = -64 / (-32)
t = 2
Therefore, it takes 2 seconds for the object to reach its maximum height.
To find the time it takes for the object to reach its maximum height, we need to determine the vertex of the parabolic equation.
The equation representing the object's height as a function of time is h = -16𝑡^2 + 64𝑡 + 80.
The vertex form of a parabola is given by h = a(t - h)^2 + k, where (h, k) represents the coordinates of the vertex.
In the equation h = -16𝑡^2 + 64𝑡 + 80, the coefficient of the t^2 term is -16, which corresponds to the value of "a" in the vertex form.
To find the vertex, we need to use the formula t = -b / (2a), where "b" is the coefficient of the t term and "a" is the coefficient of the t^2 term.
In this equation, a = -16 and b = 64. Substituting these values into the formula gives:
t = -64 / (2 * -16) = 64 / 32 = 2
Therefore, the object reaches its maximum height after 2 seconds.
To find the maximum height, we can substitute the value of t = 2 into the equation h = -16𝑡^2 + 64𝑡 + 80:
h = -16(2)^2 + 64(2) + 80 = -64 + 128 + 80 = 144
So, the maximum height reached by the object is 144 feet.