Conversion of price in different seconds to determine the speed of the fall

Question

I'm having a thought problem. I'm looking for a conversion to determine the speed of the price decrease. e.g if

\begin{align} \$398 & = 120\text{ seconds} \\ \$62 & = 0\text{ seconds} \\ ?? & = 60\text{ seconds} \\ ?? & = 1\text{ second} \end{align}

So I want to know how much the price is at 60 seconds and how much the price is at 1 seconds. Through cross multiplication, I already know that the price decrease is convex but somehow I am having problem to find a solution to this problem. Any idea?

Answer 1

I would do the following, P = (398,120), Q = (62,0)

To find the mid point is easy.

S(t) = P+t(Q-P)

S(60/120) = P + .5(Q-P)

S(1/2) = (P + Q)/2 = (230,60)

Then simply repeat the formula for 1/120

S(119/120) = P + (119/120)(Q-P)

(398,120)+(119/120)(-336,-120) = (398-333.2,120-119)=(64.8,1)

Hope that helps.

Answer 2

It looks like the price is increasing, not decreasing. The points are $(120,398)$ and $(0,62)$ If you want a straight line, you can use the two point form for the line, $p$ for price and $t$ for time, it would be $p-62=(398-62)\dfrac {t-0}{120-0}$ where I left the zeros in to show the form. Now plug $1,60$ in for $t$ and you can read off the price. Without some further information about convexity, you can't do any better.