User:Nsh/Flare-question

Problem: To find a (quadratic) equation relating the height of a flare to the time from firing, given the following information:

Height is 120 when t = 4
Height is 192 when t = 16
Maximum height occurs at t = 12

Let us form the solution in completed-square form:
$y=a(t-b)^{2}+c{\frac {}{}}$
The maximum height will occur when the first derivative is zero, ie:
$dy/dx=2a(12-b)=0\implies 12-b=0\implies b=12$

Now we have two equations with two unknows, which can be solved simultaneously.

(1) $120=a(4-12)^{2}+c{\frac {}{}}$
(2) $192=a(16-12)^{2}+c{\frac {}{}}$

(1) - (2) yields: $-72=60a\implies a={\frac {-72}{60}}=-1.2$

Putting the results back into the first equation: $120=-1.2*64+c\implies c=120--1.2*64\implies c=196.8$

This gives us the final equation as: $y=-1.2(t-12)^{2}+196.8{\frac {}{}}$