Sure! Let's break down the problem step by step.
Step 1: Understand the given information.
We are given a function h(t) = -16t^2 + 64t + 192, where h represents the height above the ground and t represents time in seconds. The coefficient -16 in front of t^2 represents the effect of gravity on the snowball.
Step 2: Find the maximum height.
To find the maximum height of the snowball, we need to determine the vertex of the parabolic function. The formula for finding the vertex of a quadratic function in the form ax^2 + bx + c is given by x = -b/2a. In our case, a = -16 and b = 64.
Using the formula, we have:
t = -64 / (2 * -16)
t = -64 / -32
t = 2
The maximum height occurs when t = 2 seconds.
Step 3: Substitute the value of t into the equation to find the maximum height.
To find the maximum height, substitute t = 2 back into the original equation h(t):
h(2) = -16(2)^2 + 64(2) + 192
h(2) = -64 + 128 + 192
h(2) = 256
So, the maximum height of the snowball above the ground is 256 feet.
Step 4: Analyze the motion of the snowball.
Since the coefficient of t^2 is negative (-16), the parabolic function is concave down. This means that the snowball will reach a maximum height at t = 2 seconds and then start to descend.
Step 5: Answer any specific questions or additional steps.
If you have any further questions or need assistance with specific parts of the problem, feel free to ask!