A ball is thrown straight up from the top of a building 148 ft. tall with an initial velocity of 72 ft per second. The distance
s(t) (in feet)
of the ball from the ground is given by
s(t) = 148 + 72t − 16t2.
Find the maximum height attained by the ball.
To find the maximum height attained by the ball, we need to find the vertex of the quadratic function represented by s(t).
The vertex form of a quadratic function is given by:
s(t) = a(t - h)^2 + k
In our case, a = -16 (because the coefficient of t^2 is -16), and we want to find the maximum value of s(t), which will be the vertex point (h, k) of the quadratic function.
To find the vertex, we can use the formula:
h = -b / (2a)
Considering s(t) = 148 + 72t - 16t^2, we have:
a = -16
b = 72
Substituting these values into the formula, we get:
h = -72 / (2 * -16)
Simplifying:
h = 72 / 32
h = 2.25
Now, let's find k by plugging in this value of h into s(t):
s(t) = 148 + 72t - 16t^2
s(2.25) = 148 + 72(2.25) - 16(2.25)^2
Simplifying:
s(2.25) = 148 + 162 - 16(5.0625)
s(2.25) = 148 + 162 - 81
s(2.25) = 229
Therefore, the maximum height attained by the ball is 229 ft.
To find the maximum height attained by the ball, we need to determine the highest point on the ball's trajectory. This occurs when the ball reaches its highest vertical position. In other words, we need to find the highest point on the graph of the equation s(t) = 148 + 72t − 16t^2.
To do this, we can use calculus. The maximum height corresponds to the vertex of the parabolic function, which can be found by determining the value of t that maximizes s(t).
The first step is to find the derivative of s(t) with respect to t:
s'(t) = 72 - 32t
Next, we set the derivative equal to zero and solve for t:
72 - 32t = 0
-32t = -72
t = 2.25
This tells us that at t = 2.25 seconds, the ball reaches its highest point.
We can now substitute this value of t into the original equation s(t) = 148 + 72t − 16t^2 to find the maximum height:
s(2.25) = 148 + 72(2.25) − 16(2.25)^2
s(2.25) = 148 + 162 - 16(5.0625)
s(2.25) = 148 + 162 - 80.9
s(2.25) ≈ 229.1
Therefore, the maximum height attained by the ball is approximately 229.1 feet.
well, the vertex is at t = -b/2a = 72/32 = 9/4
So, find s(9/4)