# Football Math

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• Kevin is a high school punter and wants to play in college but doesn't know if he hits the punts hard enough to get them out of the defense.
• Kevin asks his coach, who is also the high school math teacher, to help him compare his power and speed to that of the average college punter's time of 1.5 seconds. At this point the ball would still be in danger of being blocked by the defense.
• We can also use the instantaneous rate of change to find the speed at the average point where is usually clears and compare to the punters in college speeds
• We use the equation for gravity of f(x)=-16t^2+intail speedt+starting height (t is time)
• First, I'll time the entire punt and measure the initial power you hit it with. We will build the equation off that.
• So, we can plug that data in the gravity equation to find the distance and find the speed that is best for clearing the defense.
• The Device clocked that you hit the ball at 78ft/s and you hit it at a height of 3 ft.
• f(x)=-16x^{2}+78x+3Find Instantaneous Rate of Change at 1.5 Through f(1.5+h)-f(1.5) = 3(-16h^2+30h+84)-84=h-16h^2+30h=hh(-16h+30)=h-16h+30 and Lim-0-16*0+30=30So, the speed is 30ft/sKevins away speed is at 30 feet per second at 1.5 seconds
• Average for College punter =f(x)=-16x2} +83x+3Find Instantaneous Rate of Change at 1.5Through f(1.5+h)-f (1.5)h = (-16h^2+35h+91.5) -91.5h=-16h^2+35hh=h(16h+35)h=16h+35 And lim h-016*0+35=35 So, the Speed at 1.5 seconds is 35ft/s The average away speed for college punters is 35 feet per second at 1.5 seconds
• Ok Kevin all this shows that right now you are below average power and speed for many college punters and not getting it put of danger quick enough.
• So, the average college punters are hitting it at an extra 5 ft/s than what I hit mine at.
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