Candles

Before Christmas, Eve bought two cylindrical candles—red and green. Red was 1 cm longer than green. She lit a red candle on Christmas Day at 5:30 PM and a green candle at 7:00 PM, leaving them on fire until they burned. At 9:30 PM, both candles were the same height. Green finished at 23:00 and red at 23:30.

How tall was the red candle when it was new?

Final Answer:

c =  15 cm

Step-by-step explanation:

$$\begin{gathered} z-1=c \\ (23.50-17.50)\cdot v_1=c \\ (23.00-19.00)\cdot v_2=z \\ (21.50-17.50)\cdot v_1=(21.50-19.00)\cdot v_2 \\ c-z=-1 \\ c-6v_1=0 \\ 4v_2-z=0 \\ 4v_1-2.5v_2=0 \\ Row2-Row1\to Row2 \\ -6v_1+z=1 \\ Row4-\frac{4}{-6}\cdot Row2\to Row4 \\ -2.5v_2+0.6667z=0.6667 \\ Row4-\frac{-2.5}{4}\cdot Row3\to Row4 \\ 0.0417z=0.6667 \\ z=\frac{0.66666667}{0.04166667}=16 \\ {v}_2=\frac{0+z}{4}=\frac{0+16}{4}=4 \\ {v}_1=\frac{1-z}{-6}=\frac{1-16}{-6}=2.5 \\ c=-1+z=-1+16=15=15\text{cm} \\ c=15 \\ {v}_1=\frac{5}{2}=2.5 \\ {v}_2=4 \\ z=16 \\ \text{ Verifying Solution: } \\ {x}_1=(21.50-17.50)\cdot v_1=(21.50-17.50)\cdot 2.5=10\text{cm} \\ {x}_2=(21.50-19.00)\cdot v_2=(21.50-19.00)\cdot 4=10\text{cm} \end{gathered}$$

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