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:
The maximum height will occur when the first derivative is zero, ie:
Now we have two equations with two unknows, which can be solved simultaneously.
(1) ![{\displaystyle 120=a(4-12)^{2}+c{\frac {}{}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c14d1d2743e8357ffa2b2c16a8827936a3dca1b)
(2) ![{\displaystyle 192=a(16-12)^{2}+c{\frac {}{}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf2c7f3ae31e559d17c52ced42a125aa4ebaaf5)
(1) - (2) yields: ![{\displaystyle -72=60a\implies a={\frac {-72}{60}}=-1.2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/640bcaeb7f9bcc585d61ba210741d008dcb2f855)
Putting the results back into the first equation: ![{\displaystyle 120=-1.2*64+c\implies c=120--1.2*64\implies c=196.8}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c604163de87f0ee4bcaf76f3b8871ffcf15e7dbe)
This gives us the final equation as: