Question 147844
This is distance, {{{d}}} as a funtion of time
Here is the most general way to express it as a quadratic:
{{{d(t)  = a*t^2 + b*t + c}}}
There are {{{3}}} different snapshots of the dive, so that
should give me {{{3}}} equations, and I should be able to
find {{{a}}}, {{{b}}}, and {{{c}}}
{{{24 = a*(1)^2 + b*(1) + c}}}
{{{18 = a*(2)^2 + b*(2) + c}}}
{{{2 = a*(3)^2 + b*(3) + c}}}
Now figure out the square terms
{{{24 = a + b + c}}}
{{{18 = 4a + 2b + c}}}
{{{2 = 9a + 3b + c}}}
I'll multiply the 2nd equation by {{{3}}}, and the 3rd one by {{{2}}}
{{{54 = 12a + 6b+ 3c}}}
{{{4 = 18a + 6b + 2c}}}
Now subtract
{{{50 = -6a + c}}}
Now subtract the 1st from the 2nd
{{{-6 = 3a + b }}}
Multiply by {{{2}}}
{{{-12 = 6a + 2b}}}
Add
{{{38 = 2b + c}}}
Subtract thre 3rd one from this
{{{36 = -9a - b}}}
{{{b = -9a - 36}}}
{{{-6 = 3a + (-9a - 36)}}}
{{{-6 = 3a - 9a - 36}}}
{{{6a = -30}}}
{{{a = -5}}}
{{{50 = -6*(-5) + c}}}
{{{50 = 30 + c}}}
{{{c = 20}}}
{{{38 = 2b + 20}}}
{{{2b = 18}}}
{{{b = 9}}}
So, the equation I come up with is:
{{{d(t)  = a*t^2 + b*t + c}}}
{{{d(t) = -5t^2 + 9t + 20}}} answer
Let's see if it satisfies the data given
{{{d(1) = -5*1^2 + 9*1 + 20}}}
{{{24 = -5 + 9 + 20}}}
{{{24 = 24}}}
OK
{{{d(2) = -5*2^2 + 9*2 + 20}}}
{{{18 = -20 + 18 + 20}}}
{{{18 = 18}}}
OK
{{{d(3) = -5*3^2 + 9*3 + 20}}}
{{{2 = -45 + 27 + 20}}}
{{{2 = 2}}}
OK