# Question f28b6

May 12, 2017

The height of the candle after $6$ hours, will be $4$ in.

#### Explanation:

A candle is 10 in. tall after burning for 2 hours. After 3 hours, it is 8.5 in. tall. How tall will the candle be after 6 hours?

We can make a linear equation to model the situation. Since time is the independent variable and height is the dependent variable, we can let $\left(x , y\right)$ be $\left(\text{time", "height}\right)$, therefore the question gives us the points $\left(2 , 10\right)$ and $\left(3 , 8.5\right)$, which we can use to find the slope, m:

$m = \frac{10 - 8.5}{2 - 3} = - \frac{1.5}{1} = - \frac{3}{2}$

Consider point-slope form:
$\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)$
where $m$ is the slope of the line and $\left({x}_{1} , {y}_{1}\right)$ is a point of the line.

Now, we can write a linear equation using point-slope form:
$y - 10 = - \frac{3}{2} \left(x - 2\right)$

Since we want to know the height of the candle after 6 hours, we simply plug $x = 6$ into the above equation and solve for $y$:
$y - 10 = - \frac{3}{2} \left(6 - 2\right)$
$y = - \frac{3}{2} \left(4\right) + 10$
$y = - 6 + 10$
$y = 4$

Therefore, the height of the candle after $6$ hours, will be $4$ in.

May 12, 2017

In support of the solution submitted by 'Y'

4 inches

#### Explanation:

Sometimes it is a good idea to 'scribble' a quick sketch to assist in visualisation of what the question is asking. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Let the required height be $h$

What the target is:

Reference height - $x$ = h

where $x$ = burn rate x 4 hours
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Determine the burn rate}}$

$\left(\text{EB-EC inches")/("3-2 hours}\right) \to \frac{10 - 8.5}{3 - 2} = \frac{1.5}{1} = \frac{3}{2}$

$\text{ } \frac{3}{2}$ inches in 1 hour

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine } x}$

 x=3/2 ("inches")/("hour")xx 4" hours"

x=3/(cancel(2)^1)xxcancel(4)^2" "("inches")/(cancel("hour"))xx cancel(" hours")#
$\text{ } \textcolor{red}{\uparrow}$
$\textcolor{red}{\text{manipulating units the same way as numbers}}$

$x = 3 \times 2 \text{ inches "=" "6" inches}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine } h}$

$h = 10 - x \text{ inches "=" "10-6" "=" "4" inches}$