I'm not sure how to go about solving this problem. Please help!

Question

A large candle is 119 cm tall. It is designed to burn more quickly when it is first lit and more slowly as it approaches the bottom. Specifically, the candle takes 10 seconds to burn down the first centimeter from the top, 20 seconds to burn down the second centimeter, and 10k seconds to burn down the k-th centimeter. Suppose it takes T seconds for the candle to burn down completely. Then T/2 seconds after it is lit, the candle's height in centimeters will be h. Find 10h.

Answer 1

One way to solve this would be to think of this as an arithmetic sequence, where each term is 10 more than the previous term (that is the common difference = 10).

 

We can find the sum of all the times for each centimeter burned using the formula:  S = n/2(a₁ + a _n). In this problem, S = T, a₁ = 10 sec, and n = 119, and a_n = 10*119 = 1190 sec

 

So total time to burn the candle = T = 119/2(10 + 1190) = 71,400 sec

 

Next, we want the total time passed to be T/2, and so find the value of n, or in this problem, h, the #of cm burned after 1/2 the total time. 

 

71,400/2 = h/2(10 + 10h). So 71,400 = 10h + 10h² ,

So 7140 = h + h²   Rearranging:  h² + h - 7140 = 0

 

This factors into (h + 85)(h - 84) = 0^•   (• see below)

So h = 84 cm (the other value, -85, makes no sense here)

 

So 10h = 840 sec     (•Quick way to get factors of a big number which only differ by 1 is to take the square root of 7140 (84.5), and then try 84/85 - which was correct)