Applied exercises: Linear functions

A candle is $18\; \text{cm}$ long before being lit. When burning, it becomes shorter by $1.5\; \text{cm}$ every hour.

How many hours does it take for the candle to burn down completely?

Für diese Aufgabe benötigst Du folgendes Grundwissen: Lineare Funktionen
Each hour the candle gets shorter by $1.5\; \text{cm}$ . You get the burning time in hours by calculating how many times the $1.5\; \text{cm}$ is included in the total length of $18\; \text{cm}$ : $\frac{18}{1{,}5}=12$
Answer: After 12 hours the candle has burned down completely.
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Draw the graph of the function $f$ describing the length $y$ (in $\text{cm}$ ) of the candle in terms of the burning time $x$ (in hours).

Für diese Aufgabe benötigst Du folgendes Grundwissen: Linear function
Graph of $f$
To draw a linear function, you need two points that lie on the graph.
One point is given in the problem setting. At the beginning $x = 0$ the candle is $y = 18$ centimeters long, so $P\left(0 \vert 18\right)$ is a point on the graph.
In part a) you found out that the candle burs down after $x = 12$ hours , i.e. the candle length is then $0\; \text{cm}$ . So $Q\left(12 \vert 0\right)$ also lies on the graph.
Draw the two points in a coordinate system and connect them.
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Find the equation of $f$ .

Für diese Aufgabe benötigst Du folgendes Grundwissen: Linear function
Finding the function equation
Examples of the candle length $y$ after a certain burning time $x$ :
At the beginning, $\textcolor{006400}{y}=18$ .
After $\textcolor{ff6600}{1}$ hour, the length of the candle in $\text{cm}$ is:
$\displaystyle \textcolor{006400}{y}=18 -1.5 \cdot \textcolor{ff6600}{1}=16.5$
After $\textcolor{ff6600}{2}$ hours, the length of the candle in $\text{cm}$ is:
$\displaystyle \textcolor{006400}{y}=18-1.5\cdot\textcolor{ff6600}{2}=15$
After $\textcolor{ff6600}{x}$ hours, the length of the candle in $\text{cm}$ is:
$\displaystyle \textcolor{006400}{y} = 18-1.5\cdot \textcolor{ff6600}{x}$
So the function equation is $y=-1.5\cdot x+18$
Answer: The equation describing the length of the candle as a function of time is:
$\displaystyle y=-1.5\cdot x+18$
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Calculate the length of the candle after 5 or 8 hours. Check your calculated values using the graph.

Calculate after how many hours, only $3 \; \text{cm}$ of the candle are remaining.

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$\displaystyle y$	$=$	$\displaystyle 18-1.5\cdot\ x$
	↓	plug in $x = 5$
	$=$	$\displaystyle 18-1.5\cdot5$
	$=$	$\displaystyle 18-7.5$
	$=$	$\displaystyle 10.5$

$\displaystyle y$	$=$	$\displaystyle 18-1.5\cdot\ x$
	↓	Plug in $x = 8$
	$=$	$\displaystyle 18-1.5\cdot\ 8$
	$=$	$\displaystyle 18-12$
	$=$	$\displaystyle 6$

$\displaystyle y$	$=$	$\displaystyle 18-1.5\cdot\ x$
	↓	plug in $y = 3$
$\displaystyle 3$	$=$	$\displaystyle 18-1.5\ \cdot\ x$	$\displaystyle +1.5\ \cdot\ x$
	↓	Shift terms with $x$ to the left.
$\displaystyle 3+1.5\cdot\ x$	$=$	$\displaystyle 18$	$\displaystyle -3$
	↓	Shift numbers to the right.
$\displaystyle 1.5\ \cdot\ x$	$=$	$\displaystyle 15$	$\displaystyle :1.5$
$\displaystyle x$	$=$	$\displaystyle \frac{15}{1.5}$
	$=$	$\displaystyle 10$

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Graph of $f$

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Finding the function equation

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Plug $x = 5$ into the function equation.

Plug x=5 into the function equation.

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Plug $y = 3$ into the function equation and solve for $x$ .

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Graph of fff

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Finding the function equation

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Plug x=5x=5x=5 into the function equation.

Plug x=5 into the function equation.

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Plug y=3y = 3y=3 into the function equation and solve for xxx.

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Graph of $f$

Plug $x = 5$ into the function equation.

Plug $y = 3$ into the function equation and solve for $x$ .