Candle Question

Question

A candle is 12 inches long. In one hour it burns to 10 inches long. In 3 hours it burns to 5.5 inches. When will it burn out?

Is the height of the candle a function of time? If yes, is it a linear function?

Answer 1

in one hour it burned 2 inches

in 3 hours it burned 6.5 inches or 6.5/3 = 2.1666... inches

It's almost linear. Had it been more linear, it would have been 6 inches in 3 hours. It's instead burning a little faster as the candle gets smaller.

Had it been completely linear, it would burn out in 6 hours. Instead it will burn out a little sooner.

A common decay function is an exponential function f(x) = e^x = e^rt where r= percentage decay and t = time

but this leads to approaching zero but never reaching it, as if the candle never goes out

Another simple function that may apply is a parabola or quadratic function. Let (0,12) be the vertex of a downward opening parabola. y=ax^2 + 12, with a<0. 5.5=ax^2 + 12, -6.5 = a(3)^2, a = -6.5/9

y=(-6.5/9)x^2 + 12

set y=0 for when the candle goes out

0 =(-6.5)x^2 + 12

12 = 6.5/9)x^2

x^2 = (9/6.5)(12) = 108/6.5

x = about 4.08 hours to burn out

3<x<6, so 4.09 is reasonable time for zero candle