# A candle is 7 in tall after burning for 1 h. The same candle is 5.5 in tall after burning for 4 h. How tall will the candle be after burning for 6 h?

Apr 19, 2017

$l = \frac{15 - t}{2}$ hours
The length of the candle ($l$) after 6 hours will be 4.5 inches.

#### Explanation:

graph{x = (15 - y)/2 [-1, 10, -4.56, 16]}

We know that the length of the candle changed from 7 inches to 5.5 inches between time 1 hour and 4 hours. The rate of change can be calculated in inches per hour.

$m = \frac{\Delta h}{\Delta t} = \frac{{h}_{f} - {h}_{i}}{{t}_{f} - {t}_{i}}$
$m = \frac{5.5 - 7}{4 - 1} = \frac{1.5}{3} = - \frac{1}{2}$

The height at time 0 can be calculated by plugging in the slope $m$ into our line equation:
$y = m x + b$
We know $y$ at time 4 hours is 7 inches.
$7 = - \frac{1}{2} \cdot 1 + b$
$7 = - \frac{1}{2} + b$
$7.5 = b$

Plugging both of those values into the equation and using the variables $l$ for the length of the candle and $t$ for time we get:
$l = - \frac{1}{2} \cdot t + 7.5$

And we can calculate that at time $t = 6$ the length of the candle will be:
$l = - \frac{1}{2} \cdot 6 + 7.5$
$l = - 3 + 7.5$
$l = 4.5$

Here's a video of a candle burning over a couple of hours to inspire you:
Here's a video

Apr 19, 2017

$\textcolor{red}{\text{There is a trap in this question}}$

After 6 hours the height will be 4.5 inches

#### Explanation:

The time to reduce to the 7 inches tall is of importance. Unfortunately we do not have the starting height.

$\textcolor{b l u e}{\text{Determine the ratio of the burning rate}}$

$\left(\text{distance B to C")/("time B to C")->(7-5.5)/(4-1)color(magenta)(=1.5/3" as a ratio}\right)$

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$\textcolor{b l u e}{\text{Determine the measurement B to D } \to x}$

$\left(\text{distance B to D")/("time B to D}\right) \to \frac{x}{6 - 1} = \frac{x}{5}$

This will have the same ratio as our previously calculated burning rate giving:

$\frac{1.5}{3} \equiv \frac{x}{5}$ where $\equiv$ means equivalent to.

Matching denominators: to change 3 to 5 we multiply by $\frac{5}{3}$

So if we multiply the denominator by $\frac{5}{3}$ we have to do the same to the numerator.

$\textcolor{g r e e n}{\frac{1.5}{3} \textcolor{red}{\times 1} \text{ "->" "1.5/3color(red)(xx(color(white)(.)5/3color(white)(.))/(5/3))" "->" "2.5/5=("distance B to D")/("time B to D}}$

$\textcolor{m a \ge n t a}{\frac{x}{5} = \frac{2.5}{5} \implies x = 2.5}$

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$\textcolor{b l u e}{\text{Determine distance D to E}}$

$h = D E \text{ "=" "BE-x" " =" } 7 - 2.5 = 4.5$