Applications of linear systems

Two burning candles

Here, you can see two candles, which burn down slowly. Since the red candle is much thinner than the blue one, it will burn down faster. At the beginning, the blue candle is $6 c m$ tall, and the red one is $13 c m$ tall. By watching similar candles of the same kind, you know already that in one hour, the blue one burns down by $5 m m$ and the red one by $9 m m$ .

Write down a function equation for each of the two candles that represents its height $h$ as a function of time $t$ . So you end up with two equations.

Blue candle
You have to find the parameters of the equation $h_{b l u e} (t) = m_{b l u e} \cdot t + c_{b l u e}$ .
In one hour it burns down by $0.5 c m$ . Thus the slope is $m_{b l u e} = - 0.5$ .
The y-axis intercept is the initial height of the candle: $c_{b l u e} = 6$ .
That way, you get the equation of the function: $h_{b l u e} (t) = - 0.5 \cdot t + 6$
Blaue Kerze
$h_{r e d} (t) = m_{r e d} \cdot t + c_{r e d}$
The red candle burns down by $0.9 c m$ per hour. The slope is therefore $m_{r e d} = - 0.9$ .
The red candle initially has a height of $c_{r e d} = 13$ .
Hence, you get the function equation: $h_{r e d} (t) = m_{r e d} \cdot t + c_{r e d}$
The height of the candles is described by two linear functions.
It is convenient to use $t$ ainstead of $x$ as the variable on which others depend, on order to illustrate that the height of the candles as a function of time.
Now, compute after how many hours the candles will be the same length by turning the two functions into equations with the variables $h$ and $t$ .

For this task you need the following basic knowledge: System of linear equations
First, you create a system of linear equations with two variables from the functions:
$\begin{array}{lrll} I & h & = & - 0.5 t + 6 \\ II & h & = & - 0.9 t + 13 \end{array}$
To find the time $t$ at which both candles have the same height, you look for the solution of the system of equations. The elimination by equating coefficients is suitable, here, since both equations are solved for $h$ .
1. Equate
$\Rightarrow - 0.5 t + 6 = - 0.9 t + 13$
2. Solve for t
$\begin{aligned} - 0.5 t + 6 & = & - 0.9 t + 13 & | + 0.9 t \\ 0.4 t + 6 & = & 13 & | - 6 \\ 0.4 t & = & 7 & | : 0.4 \\ t & = & 17.5 \end{aligned}$
So the candles are the same height after $17.5$ hours, that is $17$ hours and $30$ minutes.
Answer: After $17$ hours and $30$ minutes, the candles have the same height $h$ .
The result from b) is not very realistic. Do you know, why?

For this task you need the following basic knowledge: System of linear equations
Height of the candles
According to $b)$ , the two candles have the same height after $17.5$ hours. But how tall are they?
To find this out, plug $t = 17.5$ into one of the equations from b) and calculate $h$ .
$\begin{array}{lll} h & = & - 0.5 \cdot (17.5) + 6 \\ = & - 8.75 + 6 \\ = & - 2.75 \end{array}$
So, after $17.5$ hours, both candles should be $- 2.75 c m$ tall.
What does $- 2.75$ mean?
Mathematically, the two straight lines from a) have an intersection point, there.
But in reality, this solution makes no sense. From the straight line equation of the red candle $h_{r e d} = - 0.5 t + 6$ we can calculate that it has already burned down after 12 hours. (see calculation of zeros).
Alternatively, you can draw the graphs of the candle heights, to interpret your solution.
Answer:
Although a solution arises mathematically, the two candles are never the same size: they will have burnt down, befor this may happen.
Hint: compute, to wich height $h$ both candles will have burnt down, provided that your solution for $t$ is correct.

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Applications of linear systems

Blue candle

Blaue Kerze

Height of the candles

What does −2.75 mean?

What does $- 2.75$ mean?