Doing the Math
Let's try a word problem together, shall we?
1) Let's assume we have a vengeful prankster. We will call him mj_notes. Our prankster arrives at his office to see his office cube filled with balloons and chairs, and sees a confession about said prank by one of his friends on Facebook.
He decides to retaliate by completely packing his friend's front door with snow overnight. Let's assume the door is 7 feet tall, and 3 feet wide. mj_notes wants to pack the entire door with at least a half-inch of snow. How much snow does mj_notes think that he'll need to carry out his prank?
2) During the packing process, mj_notes learns something about friction. It turns out, a slope of greater than 60 degrees leads to a collapse of the wall! If mj_notes builds a wall with a slope of 60 degrees instead, how much snow will he ACTUALLY need to carry out his diabolical plan? Assume the minimum thickness is still a half inch.
3) mj_notes finds that he can shovel about 5 cubic feet of snow an hour into his project. He starts his project at 1 AM. How high is his wall at 3 AM, when he realizes this was a really dumb idea?
4) How long would it have taken to fully complete the wall at that rate?
5 ) How long would this wall have taken if mj_notes could have used his original plan in Part 1, without worrying about the slope?
1) This is just calculating a volume. 7 feet tall * 3 feet wide * 0.5 inches thick * 1 foot/12 inches = .875 cubic feet.
2) Take 7 feet and divide by the square root of 3. That's the distance away from the door we've gotta go.
The triangle volume is: 1/2 * (7 feet tall) * (7 feet/sqrt(3) thick) * (3 feet wide) = 42.435 cubic feet
Now, add in our original wall, because we need it to be at least 1/2 inch everywhere: 43.31 cubic feet
3) Time for a little algebra.
[1/2 * x * x/sqrt(3) * 3 feet wide] + x * 3 feet wide * 0.5 inches thick * 1 foot/12 inches = 10 cubic feet
.866 * x^2 + .125 x = 10
x^2 + .144 x = 11.547
completing the square yields:
x^2 + .144 x + .005184 = 11.547 + .005184
(x+.072)^2 = 11.552
x+.072 = 3.399
x = 3.327 feet
4) We need 42.435 cubic feet in total. If we are getting 5 feet per hour, then:
5 feet / 1 hour = 42.435 feet / x hours
x hours = 1 hour * 42.435 feet / 5 feet = 8.487 hours
5) We would need .875 cubic feet instead. So:
5 feet / 1 hour = .875 feet / x hours
x hours = 1 hour * .875 feet / 5 feet = .175 hours
I learned this the hard way last night "out in the field." For the record, I got fed up at 3 AM and decided to try packing the snow at a really steep angle, hoping that it might hold. By 4 AM, I had the door covered, and I was proud. I walked to my car to get my camera, and when I came back, my hour of hard work had fallen to the ground in a mini-avalanche.
Since then, I have thought of many ways that I could have made this work. But, by the time I thought of better ideas, I was already knee-deep in a miniature sled hill. This is one of my pranks that could've gone MUCH better. ;)