Rachel has two crayons. The fat crayon burns down to nothing after 15 hours. Her skinny crayons wear down to nothing ofter 6 hours. At 10:30 she began coloring with both crayons of the same height, one fat one skinny. At what time will the skinny crayon be half the height of the fat crayon?
To calculate the time at which the skinny crayon will be half the height of the fat crayon, we first need to find out how much each crayon burns down per hour.
The fat crayon burns down to nothing after 15 hours, so it burns down 1/15th of its height each hour. Similarly, the skinny crayon wears down to nothing after 6 hours, so it burns down 1/6th of its height each hour.
At 10:30, Rachel starts coloring with both crayons of the same height. Let's assume the height of each crayon is 'h'.
After x hours, the fat crayon's height will be h - (1/15)x, and the skinny crayon's height will be h - (1/6)x.
Since we want to find the time at which the skinny crayon will be half the height of the fat crayon, we can set up the following equation:
h - (1/6)x = 0.5 * (h - (1/15)x)
To solve for x (the time at which the skinny crayon is half the height of the fat crayon), we can simplify the equation:
h - (1/6)x = 0.5h - (1/30)x
(1/30)x = 0.5h - h + (1/6)x
(1/30 - 1/6)x = -0.5h
(-1/60)x = -0.5h
x = (0.5h) / (1/60)
x = 30h
Therefore, after 30 hours, the skinny crayon will be half the height of the fat crayon. Adding this to the starting time of 10:30, we can determine the final time.
If Rachel starts coloring at 10:30, after 30 hours the time will be:
10:30 + 30 hours = 16:30 (or 4:30 PM)
So, the skinny crayon will be half the height of the fat crayon at 4:30 PM.