Problem 269

Solution 1

A 100 cm tall red candle takes 600 minutes to burn completely.

Therefore, the red candle burns at a rate of $\text{100 cm 600 min = 1 6}$ $\text{cm / min}$.

After 180 minutes, the height of the red candle will have decreased by

$\dfrac{1}{6}$ $\text{cm / min 180 min =30 cm}$.

Solution 2

A 100 cm tall red candle takes 600 minutes to burn completely.

The fraction of the red candle that burns in 180 minutes is $\dfrac{180}{600}=\dfrac{3}{10}$.

Since the candle was initially 100 cm tall, then $\text{3 10 100 cm =30}$ cm will have burned after 180 minutes.

Therefore, the height of the red candle will have decreased by 30 cm, 180 minutes after being lit.
To reach a height of 80 cm, the green candle will have decreased by $100-80=20$ cm.

Since 20 cm is $\dfrac{20}{100}=\dfrac{1}{5}$ of the candle’s original height, then it will take $\dfrac15$ of the total time

to decrease in height to this point.

Since it takes the green candle 480 minutes to burn completely, it will take $\dfrac15\times480=96$ minutes after being lit to decrease to a height of 80 cm.
Since the red candle takes 600 minutes to burn completely, 60 minutes is $\dfrac{60}{600}$ or $\dfrac{1}{10}$ of its total burning time.

Therefore after 60 minutes, the red candle will have decreased by $\dfrac{1}{10}\times 100=10$ cm in

height. That is, the red candle will be 90 cm tall after burning for 60 minutes.

Since the green candle takes 480 minutes to burn completely, 60 minutes is $\dfrac{60}{480}$ or $\dfrac{1}{8}$ of its total burning time.

Therefore after 60 minutes, the green candle will have decreased by $\dfrac{1}{8}\times 100=12.5$ cm in

height. That is, the green candle will be 87.5 cm tall after burning for 60 minutes.

The red candle will be $90-87.5=2.5$ cm taller than the green candle 60 minutes after they are lit.
Solution 1

From part (c), the green candle will decrease in height by 2.5 cm more than the red candle every 60 minutes (since their heights decrease at constant rates).

A difference of 2.5 cm in height every 60 minutes written as a fraction is $\dfrac{2.5}{60}$ cm/min., which is equivalent to $\dfrac{5}{120}=\dfrac{1}{24}$ cm/min.

That is, the green candle will decrease in height by 1 cm more than the red candle every 24 minutes after being lit.

Therefore, the red candle will be 7 cm taller than the green candle $7\times24=168$ minutes after they are lit.

Solution 2

The red candle burns at a rate of 100 cm every 600 minutes or $\frac16$ cm/min.

The green candle burns at a rate of 100 cm every 480 minutes or $\frac{5}{24}$ cm/min.

In $t$ minutes after being lit, $\frac16t$ cm of the red candle will have burned.

In $t$ minutes after being lit, $\frac{5}{24}t$ cm of the green candle will have burned.

Since both candles began with the same 100 cm height, then the red candle is 7 cm taller than the green candle when $(100-\frac16t)-(100-\frac{5}{24}t)=7$.

Simplifying this equation, we get $\frac{5}{24}t-\frac16t=7$ and by multiplying both sides by 24, $5t-4t=7\times24$, and so $t=168$.

Therefore, the red candle is 7cm taller than the green candle 168 minutes after being lit.