The path of a firework is described by the function: h(t) = -4.9t^2+ 49t+ 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 determine the maximum height of the firework, we need to find the vertex (highest point) of the quadratic function.
The formula for the vertex of a quadratic function in standard form (ax^2 + bx + c) is:
x = -b/2a
y = f(x) = ax^2 + bx + c (plug in the value of x)
In this case, the function is already in standard form:
h(t) = -4.9t^2 + 49t + 1.5
a = -4.9, b = 49, c = 1.5
Using the formula for the vertex, we get:
t = -b/2a = -49/(2*(-4.9)) = 5
h(5) = -4.9(5)^2 + 49(5) + 1.5 = 122.5
Therefore, the maximum height of the firework is 122.5 meters.
To determine the maximum height of the firework, we need to find the vertex of the quadratic function h(t). The vertex of a quadratic function in the form ax^2 + bx + c can be found using the formula t = -b / (2a). In this case, the equation is h(t) = -4.9t^2 + 49t + 1.5.
Comparing the given equation h(t) = -4.9t^2 + 49t + 1.5 to the formula ax^2 + bx + c, we have:
a = -4.9
b = 49
c = 1.5
The time at the maximum height of the firework can be calculated using the formula t = -b / (2a).
t = -(49) / (2 * -4.9)
t = -49 / -9.8
t = 5
Thus, the maximum height of the firework occurs at t = 5 seconds.
We can substitute this value of t back into the original equation h(t) = -4.9t^2 + 49t + 1.5 to find the maximum height.
h(5) = -4.9(5^2) + 49(5) + 1.5
h(5) = -4.9(25) + 245 + 1.5
h(5) = -122.5 + 245 + 1.5
h(5) = 124 + 1.5
h(5) = 125.5
Therefore, the maximum height of the firework is 125.5 meters.