The path of a firework is described by the function: h(1) = 4.97? + 491 + 1.5
where h(t) is the height of the firework, in meters, and t is the time in seconds since the launch.
Determine the maximum height of the firework. Show work
To find the maximum height of the firework, we need to find the vertex of the function.
The vertex of the function h(t) = a(t-b)^2 + c is (b,c), where b is the x-coordinate of the vertex, and c is the y-coordinate of the vertex.
In this case, h(1) = 4.97(1)^2 + 491(1) + 1.5 = 497.47 meters.
So, the vertex is (1, 497.47).
Therefore, the maximum height of the firework is 497.47 meters.
To determine the maximum height of the firework, we need to find the vertex of the parabolic function. The vertex of a quadratic function in the form h(t) = a(t - h)^2 + k can be found using the formula:
t = -b / (2a)
In the given function, h(1) = 4.97t^2 + 491t + 1.5, we can see that a = 4.97 and b = 491.
To find the value of t, substitute the values of a and b into the formula:
t = -491 / (2 * 4.97)
t = -491 / 9.94
t ≈ -49.495
Since time cannot be negative, we discard the negative value of t, and the maximum height will occur at t = 1 second.
To find the maximum height, substitute t = 1 into the function:
h(1) = 4.97(1)^2 + 491(1) + 1.5
h(1) = 4.97 + 491 + 1.5
h(1) = 497.47
Therefore, the maximum height of the firework is approximately 497.47 meters.