Question in description

Question

two candles of equal heights but different thickness are lighted. The first burn off in 4 hours and the second in 6 hours. How long after lighting them both, will the first candle be half the height of the second?

Answer 1

This was a challenging problem.

 

 Volume = pi * radius^2 * height

 

v and r are the volume and radius of the small candle

V and R are the Volume and Radius of the larger candle

 

k is the consistent rate the candles burn per hour.

Then v = 4k and V=6k as given

 

 

4k = pi*r^2* H  and 6k = pi*R^2*H with the heights the same

 

4k/(pi*r^2) = 6k/(pi*R^2)

 

 

4/r^2 = 6/R^2 <--- pi cancels; so does k

 

4/6 = r^2/R^2

 

2/3 = r^2/R^2  

 

so the square of the radii of the candles (hence their volume, since pi and H are the same) are in 2 to 3 ratio

 

Hence, 2/3 = v/V  <--- can multiply the 2/3 ratio by pi*H to get the volumes in the same proportion

 

  v = (2/3)V  <--- solves for v

 

 

So, it takes  2 hours to burn V - (2/3)V = (1/3)V

                                                          = (1/3)(6k)

                                                           = 2k

 

                            k=1 unit of volume per hour burns

 

  (2/3)V - t*k = (1/2) [ v - t*k]  where t is the time it takes for the smaller candle to be half the larger

 

 (2/3)V - t = (1/2)[ v - t]  <---- k=1 so it multiplies out

 

 (2/3)V - t = (1/2)V - (1/2)t <--- distributive

 

 (2/3)V - (1/2)V = t - (1/2)t  <---- V on one side; t on the other

 

 (1/6)V = (1/2)t  <--- combines like terms

 

 V = 3t  <--- multiplies both sides by 6

 

6k = 3t  <---- V=6k

 

6 = 3t  <---- again, k=1

 

t=2

 

=============================================

 

ok... suppose the volumes are 10 and 15, as 10/15 = 2/3

 

The first burns off in 4 hours, so thats  10/4 = 5/2 = 2.5 units of volume per hour.

  The other candle burns off in 6 hours, so 15/6 = 5/2 = 2.5 units of volume per hour.

 

in 2 hours, the first candle shall be 10 - 2*2.5 = 10 - 5 = 5

while the larger candle shall be 15 - 2.5*2 = 15 - 5 = 10 which is twice as large

 

This is a strategy on the SAT and ACT tests called "pick a number" to get the

answer quickly, but does not PROVE the result holds for ALL possible values.

Nevertheless, it supports the result found above.

 

Thanks for the challenging problem!