A ball is thrown straight up from the top of a building 128 ft tall with an initial velocity of 80 ft per second. The height

s(t) (in feet)
of the ball from the ground, at time t (in seconds), is given by
s(t) = 128 + 80t − 16t^2.
Find the maximum height attained by the ball

max height is on the axis of symmetry of the parabola ... x = -b / (2a)

tmax = -80 / (2 * -16) = 2.5 s

plug the time into the height equation to find max height

To find the maximum height attained by the ball, we need to determine the vertex of the quadratic function representing the ball's height. The vertex provides the highest point of the parabolic curve.

A quadratic function is in the form of s(t) = at^2 + bt + c, where a, b, and c are coefficients.

In this case, a = -16, b = 80, and c = 128.

The x-coordinate of the vertex can be found using the formula -b/2a. In this case, it is -80 / (2*(-16)) = -80 / (-32) = 2.5 seconds.

To find the corresponding y-coordinate (maximum height), we plug the x-coordinate into the original equation:

s(t) = 128 + 80t - 16t^2
s(2.5) = 128 + 80(2.5) - 16(2.5)^2
s(2.5) = 128 + 200 - 16(6.25)
s(2.5) = 128 + 200 - 100
s(2.5) = 328 - 100
s(2.5) = 228 ft

Therefore, the maximum height attained by the ball is 228 feet.

To find the maximum height attained by the ball, we need to determine the vertex of the parabolic equation that represents the height of the ball over time.

The equation for the height of the ball is given by s(t) = 128 + 80t - 16t^2, where s(t) represents the height at time t.

To find the vertex, we can use the formula x = -b / (2a) to find the time at which the ball reaches its maximum height. In this equation, a, b, and c correspond to the coefficients of the quadratic equation.

In this case, a = -16, b = 80, and c = 128.

Using the formula, we can calculate the time at which the ball reaches its maximum height:

t = -80 / (2 * -16)
t = -80 / -32
t = 2.5 seconds

Now, substitute this value of t back into the equation s(t) to find the maximum height of the ball:

s(t) = 128 + 80 * 2.5 - 16 * (2.5)^2
s(t) = 128 + 200 - 16 * 6.25
s(t) = 328 - 100
s(t) = 228 ft

Therefore, the maximum height attained by the ball is 228 feet.