At a high school sports game, the principle wants to shoot fireworks over the crowd. The fireworks will be shot on top of a 160 ft tower at a 65 degree angle. The initial velocity is 92 ft./sec. In this problem we were given 2 equations: h(t) = 160 + 92t - 16t^2 (The standard form equation for our parabola) and d(t) = 92t / tan65 (Used to calculate horizontal distance)
The principle needs to know two things: Where will the fireworks land? When will the fireworks reach their peak?
None of us knew how to solve this problem, so instead we asked questions: What are the different parts of a parabola? How can we use different pieces of information to fill in the blanks for any parabola?
To do this problem, we started out by looking at parabolas and making observations. Because we had no prior experience, this led to some basic x/y tables but not much else.
Before solving
Through lots of exploration over months of working on this unit, we eventually arrived with about 5 ways of showing a parabola. By plugging in 0 for different variables, 3,4, and 5 can be proven.:
1) The actual graphed parabola on an x/y plane
2) Data and x/y tables
3) Standard form or f(x) = ax^2 + bx + c Helps to find y intercept (c). When factored (what numbers multiply to get c that add to get b?) can give you factored form. If cannot be factored, gives enough information to use the quadratic formula. If you complete the square you can get vertex form which looks like: a(x^2 + (b/2)^2)) + c - (b/2)^2 and then simplifies to: a(x+ (b/2))^2 + c - (b/2)^2
4) Factored form or f(x) = (x+(c/b))(x+(c/a)) Helps to find x intercepts (take each parentheses and make it = 0, solve algebraically). When distributed gives standard form. Can also sometimes appear as (x+(√c))^2. To get vertex form use standard.
5) Vertex form or f(x) = a(x+(b/2))^2 + c - (b/2)^2 Helps to find vertex (for x of vertex: take parentheses and make it equal 0, solve algebraically) (for y of vertex: the last term being subtracted) To get standard form work backwards: multiply (b/2) by 2, put a in front of x and square, remove last term and parentheses. To get factored form use standard.
We also discovered the corresponding parts of a parabola: 1) Y intercept: The place where the parabola crosses the y axis (0,y) 2) X intercept(s): The place where the parabola crosses the x axis (x,0) 3) Line of symmetry (LoS): The place where the parabola can be split in half and stay symmetrical 4) Vertex: The minimum (concave up) or maximum (concave down) of the parabola where the x marks the LoS.
Other terms: 1) Concave up: When the x is negative and the parabola opens up 2) Concave down: When the x is positive and the parabola opens down 3) Quadratic formula: The equation used to find the definitive x intercepts (the proof is long)
Solving the firework problem
Now having all of this information on the subject, I was able to solve the initial questions very swiftly:
Total distance is a non-mathematical way of saying "What's the distance from 0 and your x intercept. So, to find the total distance that the firework traveled, I needed to know the x intercepts. Because the fireworks were shot off of a 160 ft building (which makes 160 ft the y intercept) one of the x intercepts would be a negative number. In the real world, the firework isn't really being shot until the y intercept and the negative x intercept is just the beginning of the path it would have taken from the floor. Therefore, I only needed to find the positive x intercept and then use the second equation given to convert the x intercept from seconds into feet.
When will the fireworks reach their peak?
For this question, peak is just a non-mathematical way of saying vertex. To find the vertex, I could have converted the standard form into vertex form, but, because I already had the x intercepts in seconds, I decided to take the easier route. If you find the average of the x intercepts, you can find the number directly between them. This number is important because it is the line of symmetry, the place where two points of the parabola with different x intercepts have the same y intercept. Because the vertex is our peak, this number is also the x of our vertex. By taking our new x value and plugging it into the standard form, we get the y value, which is not necessary for the question asked but can help us make a more accurate graph.
Final answers: The firework will travel 307.4 feet total The firework will explode at it's peak after 2.875 seconds
Problem Evaluation
I really liked this problem because I felt challenged and got to explore an entire subject of math in extreme depth. The problem in itself was very easy once I had the background information that I needed. The only thing I had trouble with was not knowing what the second equation meant. It felt weird spending so long learning material in class that in the end didn't give me a 100% understanding of my final answer. Other than that I loved it!
Self Evaluation
I'm happy with my work for this problem. I have been very engaged and thoroughly challenged. Although there were times when I felt I fully understood everything and wanted to move on, it made me solidify my understanding even more by putting me in the place to teach my peers. In the next problem, I hope to take on even more challenges and be an even better group member to my peers who may be having a harder time with math.