6 hour have to change to minute. i hate words problem.
so here the problem:
Two candles are the same length. One candle takes 6 hours to burn all the way down while the other candle takes 9 hours. If the two candles are lit at the same time, how long will it take for the two candles to burn so that one candle is twice the length of the other?
Word problems are used to see whether you understand the algebraic concepts. Anyone can solve an equation. But if you encounter a problem and can then set up the equations to solve, that shows a deeper understanding.
So, guess what? In real life, no one hands you an equation; instead, you get a problem to solve.
So, here goes:
Let the candles start at height h. The units are not given, so I'll assume inches just to have a reference.
If candle x takes 6 hours to burn down, it burns at h/6 in/hr. So, at time t minutes, its height is
x = h - h/360 * t
The other candle, burning down at h/9 in/hr,
y = h - h/540 * t
When is the slow burner twice as tall as the fast burner?
h - h/9*t = 2(h - h/6*t)
you can divide all the h's out, giving
1 - t/9 = 2(1 - t/6)
1 - t/9 = 2 - t/3
2t/9 = 1
t = 9/2
So, after 4.5 hours,
x = h/2 (burned halfway)
y = t/4 (half as tall as x)
Oops cut/paste error. Left some units in minutes, but the final equations are the solution.
To solve this problem, let's break it down step by step:
Step 1: Convert the given hours to minutes.
- We are given that one candle takes 6 hours to burn completely. To convert this to minutes, we multiply 6 by 60 since there are 60 minutes in an hour. So, one candle takes 6 x 60 = 360 minutes to burn completely.
- Similarly, the other candle takes 9 hours to burn completely. Converting this to minutes, we have 9 x 60 = 540 minutes.
Step 2: Determine the common time for both candles to burn so that one candle is twice the length of the other.
- Let's assume the common time it takes for both candles to burn is "x" minutes.
- According to the problem, when both candles are lit at the same time, the shorter candle needs to be half the length of the longer candle.
- Therefore, the shorter candle will take x minutes to burn completely, and the longer candle will take 2x minutes to burn completely.
Step 3: Set up an equation based on the given information.
- Since the length of the candles is directly proportional to the time it takes to burn, we can set up the following equation: x = 360 and 2x = 540.
- Solving these equations will give us the value of x.
Step 4: Solve the equation.
- From the first equation, x = 360.
- Plugging this value into the second equation, we have 2(360) = 540.
- Simplifying further, 720 = 540.
- Since this equation is not true, it means that our assumption that both candles burn at the same time is incorrect. Therefore, it is not possible for one candle to be twice the length of the other when they both burn at the same time.
Hence, based on this analysis, we can conclude that there is no solution to the problem as stated.