A highschool soccer team has just won the championship for their school. The director has assigned you to coordinate an event to celebrate their win. You decide to have a fireworks display, but in order to finalize this, you need to create build plans. The director isn't concered about cost, but he worries about safety. The fireworks will be launched from the top of the school tower and he needs your help finding the height the fireworks will reach and the distance they will travel.
Process-
To begin this problem, my group and I came up with a series of questions we could ask to help with our understanding of how this problem worked. Some of these questions included: -How tall was the tower? -How big is area? -How many fireworks are there? -At what degree the fireworks will be launched at
Answering these questions helped us form somewhat of an image of what we needed to solve to complete the problem.
From this we decided that if we create and xy table, we could figure out at what time the fireworks would reach its vertex.
From the table, we saw that the max height was at 3 seconds, but when we solved for the height at 2.5 seconds, the height was higher than it was at 3 seconds. With this we concluded that the time the fireworks reached its vertex would be between 2 and 3 seconds. But there are an infinite amount of numbers in between 2 and 3, so we were stumped. We had to take a step back to be able to know how to solve this problem.
For the next couple of weeks, we worked on solving quadratic equations. We started multiplying binomials and from there we went to factored intercepts, then we combined them and started solving for the special points on a parabola. In order to do this, we needed to learn the equations to find the special points. First we started by looking at the problems standard forms and what they have in common with their intercepts. We noticed that the constant of the equation was the same as the y intercepts, and when the equation was in factored form, the x intercepts were the same. The most important equation we needed to learn was how to solve in vertex form. With vertex form, we could get the equation in standard for to get us the y intercept, factored form for the x intercepts, and vertex form for the vertex.
We then learned about the quadratic formula, which would be our next and final step to finally completeing the fireworks problem. Originally, we learned that if you couldn't get the factored form of an equation to equal 0, then that equation would be deemed unfactorable.
But even with the equation being unfactorable, there were still roots on the parabola. The quadratic equation was going to help us solve for those roots. Once we learned and mastered the quadratic equation, we were able to start on the fireworks problem and were going to solve it with the formulas we learned.
We went back to the orignal 2 functions that we needed to solve the problem which were, h(t)=160+92t-16t^2 and d(t)=92t/tan65. The first equation would help us find the max height the fireworks will reach, and the 2nd equation will get us the distance the fireworks will travel. But the one thing that was missing was the time this all was happening. In order to find the time part of the vertex, we would need to insert the equation into the quadratic formula.
When we inserted the equation into the formula, we got the time slots for when everything happened. At 2.88 seconds, the fireworks will reach their max height, and at 7.15 seconds, the fireworks landed. But to be more precise, the time was 7.149 seconds.
We then inserted each time period into their equations and for the vertex we got, at 2.88 seconds, the fireworks will reach their max height, 292.32 feet and at 7.149 seconds the fireworks would hit the ground, traveling a total of 306.69 feet. And we were finally done with our fireworks problem. We were able to finally complete our graph. (right:Height of vertex, left: distance traveled, bottom: completed graph)
Problem evaluation-
This problem was one of the my most proudest problem to have finally solved. Not because of the problem itself, but what the problem represented. This took 12 weeks of brain killing work that actually made me enjoy math again, the challenging equations and new concepts we were learning kept my drive going and made me want to learn more.
self evaluation-
I think that I did very good on this unit and it was the unit that I worked the hardest I ever have in. And I think its represented in my complete understanding of the problem. I think it's because of these reasons that I actually give myself an A for this unit. I'm usually not one to think highly of myself, but I truly think that I worked hard enough to deserve that A.