a Substitute 0 for t to get the firework's height during the launch — when it is on the platform.
B
b Substitute 0 for h and find the positive solution to resulting equation.
C
c By rewriting the function into graphing form we can identify the vertex.
A
a 11.27 feet
B
b 10.23 seconds
C
c 133.77feet
Practice makes perfect
a To get the height of the platform, we should look at the height of the firework during the launch. Since it launches at time t =0, we should substitute 0 for t in the given formula for the height.
This means that the platform was 11.27 feet above ground.
b Notice that if the height h is 0, it means that the firework is at ground level. Therefore, to calculate the time it takes for the firework to hit the ground we should substitute 0 for h in the given formula and solve the resulting equation for t. The smallest positive solution will be our time.
h=-4.9t^2+49t+11.27 ⇔
0=-4.9t^2+49t+11.27
Notice that we have a quadratic equation for t, written in standard form. We can easily recognize the coefficients a, b, and c, which we can then use to find the solutions using the Quadratic Formula.
0=at^2+bt+c ⇔ t=- b±sqrt(b^2-4ac)/2a
In our case we have a = -4.9, b = 49, and c = 11.27. Let's substitute these values into the Quadratic Formula.
We can simplify this result into two separate roots.
t=-49 ± 51.2044138722 .../-9.8
t_1=-49 + 51.2044138722 .../-9.8
t_2=-49 - 51.2044138722 .../-9.8
t_1=2.20441387224/-9.8
t_2=-100.204413872/-9.8
t_1=-0.224940191045...
t_2=10.224940191...
t_1≈-0.23
t_2≈10.23
Notice that one of the solutions is negative and one is positive. Since our function is supposed to measure the firework's height after the launch, we are only interested in the positive solution. Therefore, it is going to hit ground after approximately 10.23 seconds.
c Since the formula for the firework's height is a quadratic function, the maximum height will be achieved at its vertex. To find the vertex, we should rewrite the function into graphing form.
Graphing form:& y=a(x- h)^2+ k
Vertex:& ( h, k)To do that we have to complete the square. However, this requires the squared variable to have a coefficient of 1. Therefore, we will first divide both sides of the equation by - 4.9.
h=- 4.9t^2+49t+11.27
⇓
h/- 4.9=t^2-10t-2.3
Now we can complete the square and then write the function in graphing form.
Having rewritten the equation in graphing form, we can identify its vertex as (5,133.77).
Note the second coordinate of the vertex, 133.77, which gives us the maximal height of the firework. The first one tells us after how many seconds this height will be reached.
