Question

1. a shot - putter throws a shot and its height is noted every 20 feet, the data is in the table below:

feet traveled
20 40 60 80 100 120 140 160 180 200
feet above ground
25 40 55 65 71 77 77 75 71 64
what is the quadratic equation of best fit for these data?

Explanation:

Step1: Let \(x\) be the feet traveled and \(y\) be the feet above ground.

We assume the quadratic equation is of the form \(y = ax^{2}+bx + c\).

Step2: Substitute the points \((20,25)\), \((40,40)\), \((60,55)\) into the equation.

For \((20,25)\): \(25=a\times(20)^{2}+b\times20 + c=400a + 20b + c\)
For \((40,40)\): \(40=a\times(40)^{2}+b\times40 + c = 1600a+40b + c\)
For \((60,55)\): \(55=a\times(60)^{2}+b\times60 + c=3600a+60b + c\)

Step3: Subtract the first equation from the second.

\((1600a + 40b + c)-(400a + 20b + c)=40 - 25\)
\(1200a+20b = 15\) (Equation 1)

Step4: Subtract the second equation from the third.

\((3600a+60b + c)-(1600a + 40b + c)=55 - 40\)
\(2000a+20b=15\) (Equation 2)

Step5: Subtract Equation 1 from Equation 2.

\((2000a+20b)-(1200a + 20b)=15 - 15\)
\(800a=0\), so \(a =-\frac{1}{160}\)

Step6: Substitute \(a =-\frac{1}{160}\) into Equation 1.

\(1200\times(-\frac{1}{160})+20b = 15\)
\(-7.5+20b = 15\)
\(20b=22.5\), \(b=\frac{9}{8}\)

Step7: Substitute \(a =-\frac{1}{160}\) and \(b=\frac{9}{8}\) into \(400a + 20b + c=25\)

\(400\times(-\frac{1}{160})+20\times\frac{9}{8}+c=25\)
\(- 2.5+\frac{45}{2}+c=25\)
\(-2.5 + 22.5+c=25\)
\(c = 5\)

Answer:

\(y=-\frac{1}{160}x^{2}+\frac{9}{8}x + 5\)