Math Problem Statement
S. had two candles, one of which was 36cm longer than the other. He lit the longer one at 4 p.m. and lit the shorter one at 8 p.m. At 10 p.m., they were both the same length. The longer one was completely burned out at 11 p.m. and the shorter one completely burned at 1 a.m. The two candles burned at different, but constant, rates. What was the sum of the original lengths of the two candles?
Solution
Let's solve this problem step by step.
Step 1: Defining Variables
Let:
- be the length of the longer candle (in cm),
- be the length of the shorter candle (in cm),
- be the burn rate of the longer candle (in cm per hour),
- be the burn rate of the shorter candle (in cm per hour).
From the problem, we know that:
- (the longer candle is 36 cm longer than the shorter candle).
- The longer candle was lit at 4 p.m. and completely burned out at 11 p.m., so it burned for 7 hours. Thus:
- The shorter candle was lit at 8 p.m. and completely burned out at 1 a.m., so it burned for 5 hours. Thus:
Step 2: Setting up the Lengths at 10 p.m.
At 10 p.m., both candles had the same remaining length. Let’s calculate the lengths of the candles at that time:
- The longer candle had been burning for 6 hours (from 4 p.m. to 10 p.m.), so its remaining length at 10 p.m. was:
- The shorter candle had been burning for 2 hours (from 8 p.m. to 10 p.m.), so its remaining length at 10 p.m. was: Since the lengths are the same at 10 p.m., we have:
Step 3: Solving the System of Equations
We now have the following system of equations:
Substitute and into the system. This gives: Using , substitute into the above equation: Expanding and simplifying: Now substitute into this equation: Thus:
Step 4: Finding the Lengths of the Candles
Now that we have , we can find and :
Step 5: Finding the Sum of the Lengths
The sum of the original lengths of the two candles is:
Thus, the sum of the original lengths of the two candles is 58.5 cm.
Would you like a more detailed explanation, or do you have any questions?
Here are 5 related questions you could explore:
- What if the longer candle burned out at 12 a.m.? How would that change the solution?
- How does the burn rate impact the total burn time if both candles had the same initial length?
- What if both candles were lit at the same time? How long would they take to burn out?
- How would the problem change if the candles burned at variable rates over time?
- What would happen if a third candle, burning slower than the others, was introduced?
Tip: Always define your variables clearly when solving a word problem involving multiple unknowns.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Linear equations
Rate problems
Formulas
L1 = L2 + 36
L1 = 7r1
L2 = 5r2
r1 = L2 - 2r2
Theorems
-
Suitable Grade Level
Grades 9-12
Related Recommendation
Solving Candle Height Problem Using Linear Equations
Two Candles Burning at Different Rates: Finding When They Reach the Same Height
Solving Linear Equations with Candle Height and Burn Rates
Determine Candle Burn Rate Using Linear Relationships
Calculate Burn Time for a 21 cm Tall Candle Using Proportion