I have done two parts but no idea how to the third one.
The height, h, in meters, of a firecracker fired into the air can be modeled using the function h(t)=-5(x-4)^2+85, where t is the time, in seconds, after the firecracker is fired.
c) Use the graph to estimate
i) how long the firecracker is in the air
ii) the maximum height reached by the firecracker, and how long it takes to reach this maximum height
the time in the air is t when h=0 again. So, just solve for h(t) = 0.
the max height is the vertex of the parabola, clearly when t=4.
To estimate the answers, we need to use the given function for the height of the firecracker and analyze it.
The given function is h(t) = -5(t-4)^2 + 85.
i) To estimate how long the firecracker is in the air, we need to determine the time when the height becomes zero. In other words, we need to find the value of t when h(t) = 0.
Setting h(t) = 0, we get:
-5(t-4)^2 + 85 = 0.
To solve for t, let's simplify the equation:
-5(t-4)^2 = -85.
Now, let's divide both sides of the equation by -5:
(t-4)^2 = 17.
Taking the square root of both sides, we obtain two possible solutions:
t - 4 = √17 or t - 4 = -√17.
Solving for t, we get:
t = 4 + √17 or t = 4 - √17.
Since time cannot be negative in this context, the firecracker is in the air for approximately t = 4 + √17 seconds.
ii) To estimate the maximum height reached by the firecracker, we need to identify the vertex of the parabolic graph represented by the function. The vertex represents the maximum or minimum point of the parabola.
The given function is in the form h(t) = a(t-h)^2 + k, where (h, k) represents the vertex. Comparing it with the given function -5(t-4)^2 + 85, we can see that the vertex is located at (4, 85).
Therefore, the maximum height reached by the firecracker is approximately 85 meters, and it takes approximately 4 seconds to reach this maximum height.
Please note that these estimates are based on interpreting the graph and may not be entirely accurate. For precise values, you would need to solve the equations algebraically or use more precise graphing methods.