Question

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?

Answer 1

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 `(2,5) and (8,3)`.

The slope is easy to find
`(l_1-l_2)/(t_1-t_2) = (3-5)/(8-2) = -2/6 = -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(t-t_1)`

With a little algebra we can show that
`l = -1/3 t + 17/3`

And the length at time `t=0` can be read off easily:
`l = 17/3 = 5 2/3`

A quick sanity check... since `5 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 `1/3` inch per minute?