Final answer:
After approximately 26.67 hours, the two candles, one 10 inches tall burning at 1/4 inch per hour, and the other 14 inches tall burning at 2/5 inch per hour, will be the same height. We solve this by setting up an equation for each candle's height as a function of time and finding when they are equal.
Step-by-step explanation:
The question asks after how many hours two candles, initially of different heights and burning at different rates, will be the same height. To find this, we will use algebraic equations to express the height of each candle after a certain number of hours and then solve for the time at which the heights are equal.
Let's let h represent the number of hours after which the candles will be of the same height. The first candle starts at 10 inches and burns at a rate of 1/4 inch per hour. The second candle starts at 14 inches and burns at a rate of 2/5 inch per hour.
Height of the first candle after h hours: 10 - (1/4)h
Height of the second candle after h hours: 14 - (2/5)h
We set these two expressions equal to find the time when both heights are the same:
10 - (1/4)h = 14 - (2/5)h
To solve this, we combine like terms and isolate h:
(1/4)h - (2/5)h = 14 - 10
(5/20)h - (8/20)h = 4
(-3/20)h = 4
h = -4 / (-3/20)
h = -4 * (-20/3)
h = 80/3
h = 26.67
So, after approximately 26.67 hours, both candles will be the same height.