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: $$ dy/dx = 2 a (12 - b) = 0 \implies 12 - b = 0 \implies b = 12 $$

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: $$ y = -1.2 (t - 12)^2 + 196.8 \frac{}{}$$