The varsity soccer team at Jefferson High has just won the championship. To celebrate the school wants to put on a firework display. The fireworks will launch off from the top of the 160 ft tower. Our job is to figure out the time mechanism that detonates the fireworks and also make sure the viewers are safe.
Process & solution:
The knowns we were given was: 65 degree angle from the top of the tower. Y- intercept is (0,160). Initial velocity of 92 ft./sec. Tower is 160 ft. We are trying to figure out the “unknown peak” of the fireworks, aka the vertex, and the distance it will travel.
To find out the peak of the fireworks I used the equation we were given:
h(t)=160+922-16t2 In standard form:ax^2+bx+c
Quadratic formula:-b±b2-4ac 2a
I remembered that using the quadratic formula you can find the x-intercepts. So I wrote out my “ABCs” and used to quadratic formula to solve.
A: -16 -(92)±√(92)2-4(-16)(160) / 2(-16)
B: 92 -92±√8,464+10,240 / -32
C: 160 -92±√18,704 / -32
-92±136.8 / -32
-92+136.8 / -32 -92-136.8 /-32
And then once I solved the quadratic formula, I got these as my X-intercepts.
X: (-1.4,0) X: (7.15,0)
To find the “peak” of the fireworks, aka the vertex I added my two x-intercepts together then divided by 2. Then I took my answer and plugged it into h(t)=160+92t-16t
To find the horizontal distance I took my x-intercept of (0,7.15)
and plugged it into d(t)=92(t)/tan65.
I did really enjoy this problem. I liked how we got the problem first and then learned about the different equations. I didn't feel really challenged though. At first of course I didn't know how to get started but then once I got the equations I needed it was very easy. I did like the equations and steps we learned because I know that I will use that in the next few years of me doing math.
If I were to grade myself it would be an A. I chose to give myself an A because I did finish the problem and I am able to understand and teach others what I've learned. I didn't give myself an A+ because I didn't push myselfreally that hard. If It was more challenging for me and I made sense of it at the end, I would have gave myself an A+.