AlGeBrA PROBLEM! 10 POINTS!! VERY HARD!!?
A room has a floor that is 10 feet by 38 feet and a ceiling that is 20 feet high. An ant sits one foot above the floor in the middle of one of the small end walls and desires a crumb that is one foot below the ceiling in the middle of the opposite end wall. The ant crawls along the shortest possible path to reach the crumb. How far does the ant crawl?
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The ant crawls exactly 58 feet.
The shortest distance to the floor is a line exactly perpendicular to the floor. This line is one foot long, because we know that the ant is one foot off the floor.
Now, the shortest distance from that end of the floor to the other end directly below the crumb is a straight line exactly perpendicular to both walls. Because the ant is on a small wall, the distance between the two small walls is 38 feet, the length of the longer walls. So far, the ant has crawled 39 feet.
The last leg of the crawl is straight up the wall to the crumb, which is 19 feet, as the crumb is one foot lower than the 20-foot ceiling. This ends the trip at 58 feet.
You might contest that the ant could crawl up to the ceiling, across the ceiling, and down, but that would be 19 feet, then 38 feet, then one foot, which adds up to 58 feet. This is because the two paths are exact rotations of each other.
NOTE: Going diagonal is SHORTER than goiing down and across.
Draw the diagram. It's hard to draw on a computer but I drew it myself on a piece of paper. It is a triangle and you are finding the hypotenuse.
Let the hypotenuse be x.
The longer side (from one end of the floor to the other) is 38 feet.
Because the ant is 1 foot off the ground and 1 foot below the ceiling it is 12 feet below the ceiling so instead of the height of the ceiling being 20 feet, it is (20-2)feet. which is 18 feet.
Use pythagoras' theorem to find the hypotenuse.
c²=a²+b²
x²=18²+38²
x²=324+1444
x²=1768
x=â1768
x=42.04
x=42feet
So, the ant had to crawl 42 feet to reach the crumb.
I f you unfold the box and find the hypotenuse of the right triangle with legs 5 + 38 + 1 =44 and 19+5 =24, you'll get 50.12 feet. Hard to explain without a model.