A ball is thrown straight up from the top of a building 144 ft tall with an initial velocity of 48 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) = 144 + 48t − 16t2.
Find the maximum height attained by the ball.
as with all quadratics (ax^2+bx+c), the vertex is at x = -b/2a
To find the maximum height attained by the ball, we need to determine the vertex of the quadratic function s(t) = 144 + 48t - 16t^2. The vertex represents the highest point of the ball's trajectory.
Step 1: Write the function in vertex form
The vertex form of a quadratic function is given by s(t) = a(t-h)^2 + k, where (h, k) is the vertex of the parabola. We can rewrite the given function as follows:
s(t) = -16t^2 + 48t + 144
Step 2: Determine the x-coordinate of the vertex
The x-coordinate of the vertex can be found using the formula h = -b/2a. In this case, a = -16 and b = 48:
h = -48 / (2 * -16)
h = -48 / -32
h = 1.5
Step 3: Find the y-coordinate of the vertex
Substitute the value of h (1.5) into the function s(t):
s(1.5) = -16(1.5)^2 + 48(1.5) + 144
s(1.5) = -16(2.25) + 72 + 144
s(1.5) = -36 + 72 + 144
s(1.5) = 180
Step 4: Determine the maximum height
The maximum height attained by the ball is represented by the y-coordinate of the vertex, which we found to be 180 feet. Therefore, the maximum height attained by the ball is 180 feet.
To find the maximum height attained by the ball, we need to find the vertex of the quadratic equation given by the height function s(t) = 144 + 48t - 16t^2.
The vertex of a quadratic function is given by the formula (-b/2a, f(-b/2a)), where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c.
In this case, a = -16, b = 48, and c = 144. Therefore, the x-coordinate of the vertex is given by -b/2a = -48 / (2 * -16) = -48 / -32 = 1.5 seconds.
To find the y-coordinate of the vertex, we substitute the x-coordinate (t = 1.5) into the height function s(t):
s(1.5) = 144 + 48(1.5) - 16(1.5)^2
= 144 + 72 - 16(2.25)
= 144 + 72 - 36
= 180 feet.
Therefore, the maximum height attained by the ball is 180 feet.