Candle Burning Time Calculation: Uniform Rate and Length Comparison

When dealing with candles of different materials that burn at uniform rates, we often encounter interesting mathematical problems. This article explores a specific scenario involving two candles of the same length but differing burn rates. The goal is to determine the appropriate time to light the candles so that one stub is twice the length of the other at a given time. We will solve this problem step-by-step using a logical and systematic approach.

Problem Statement

The problem states that two candles have the same length but different burn rates. Candle A burns out completely in 3 hours, while Candle B takes 4 hours to do the same. We need to find the time at which these candles should be lit so that at 4 PM, one stub is exactly twice the length of the other.

Candle Burn Rates

To solve this problem, we first need to understand the burn rates of the candles.

Candle A

Candle A burns completely in 3 hours, so its burn rate is:

frac{L}{3} per hour

Candle B

Candle B burns completely in 4 hours, so its burn rate is:

frac{L}{4} per hour

Length Calculations

Let t be the time in hours since the candles were lit until 4 PM. We can calculate the remaining lengths of the candles at any given time t.

Remaining Length of Candle A

The remaining length of Candle A after t hours is:

L_A L - left(frac{L}{3}right) t L left(1 - frac{t}{3}right)

Remaining Length of Candle B

The remaining length of Candle B after t hours is:

L_B L - left(frac{L}{4}right) t L left(1 - frac{t}{4}right)

Condition for Lengths

We want L_A 2L_B at 4 PM. This gives us the equation:

L left(1 - frac{t}{3}right) 2L left(1 - frac{t}{4}right)

Dividing both sides by L (assuming L eq 0):

1 - frac{t}{3} 2 left(1 - frac{t}{4}right)

Expanding the right side:

1 - frac{t}{3} 2 - frac{t}{2}

Rearranging terms:

-frac{t}{3} frac{t}{2} 2 - 1

frac{t}{6} 1

t 6

Conclusion

Hence, the candles should be lit 6 hours before 4 PM, which means they should be lit at 10 AM.

Another Example

For another example, if the length of each candle is 12 meters (m) and the burn rates are 4 m/hr and 3 m/hr, respectively, we can calculate the time at which one candle is twice the length of the other.

Given Information

text{Length of each candle} 12 m

text{Burn rate of Candle A} 4 m/hr

text{Burn rate of Candle B} 3 m/hr

text{At} t text{hours}, one candle is twice the length of the other.

Equation

text{For Candle A}:

text{current_length_candle_a} 12 - 4t

text{For Candle B}:

text{current_length_candle_b} 12 - 3t

text{Given that}

text{current_length_candle_a} 2 times text{current_length_candle_b}

text{Substituting the formulas}:

text{12 - 4t} 2 times (12 - 3t)

text{Solving for } t:

12 - 4t 24 - 6t

2t 12

text{t} frac{12}{2} 6 text{ hours}

text{Time the candles should be lit} 4 PM - 6 text{ hours} 10 AM

Additional Tips

Given the uniform burning rates, the length of a candle can be calculated as:

text{current_length} text{max_length} - frac{text{time_in_hours}}{text{hours_to_burn}}

Example Calculation

Assume the starting length of both candles is 1:

text{current_length_candle_1} 1 - frac{text{hours}}{4}

To find the time it takes for one candle to be half the length of the other:

text{current_length_candle_a} 2 times text{current_length_candle_b}

By substituting the formulas, you can determine the number of hours it takes for one candle to reach half the length of the other.

Try solving similar problems with different starting lengths or conditions to enhance your understanding.

Conclusion

By understanding the burn rates and applying basic algebraic techniques, we can solve the problem of determining the appropriate time to light two candles of different materials so that one stub is twice the length of the other at a given time. This method can be applied to various scenarios, making it a valuable skill in mathematics and real-life problem-solving.