# What was the original length of the candle if a candle is burning at a linear rate and the candle measures five inches two minutes after it was lit and it measures three inches eight minutes after it was lit?

Dec 18, 2014

This is a problem where you are given two positions on a graph and are asked to find the equation of the line connecting them. In this case you are also being asked for the $y$ intercept of the graph.

The independent variable is time. We'll plot that on the x-axis. The dependent variable is the length of the candle. That will be the y-axis.

At two minutes the length of the candle is 5 inches.
$t = 2 , l = 5$
At eight minutes the length of the candle is 3 inches.
$t = 8 , l = 3$

We need to find the equation of a line which goes through the two points $\left(2 , 5\right) \mathmr{and} \left(8 , 3\right)$.

The slope is easy to find
$\frac{{l}_{1} - {l}_{2}}{{t}_{1} - {t}_{2}} = \frac{3 - 5}{8 - 2} = - \frac{2}{6} = - \frac{1}{3}$

The intercept can be found from the point-slope formula by inserting one of the data points into the equation:
$l - {l}_{1} = m \left(t - {t}_{1}\right)$

With a little algebra we can show that
$l = - \frac{1}{3} t + \frac{17}{3}$

And the length at time $t = 0$ can be read off easily:
$l = \frac{17}{3} = 5 \frac{2}{3}$

A quick sanity check... since $5 \frac{2}{3}$ is larger than 5 (the length at 2 minutes) the answer makes sense.

What sort of candle do you think would burn at a rate of $\frac{1}{3}$ inch per minute?