Here are two famous general formulas: 1 + 2 + 3 + ... + n = n(n+1)/2 and 12 + 22 + 32 +..+N2 = n(n + 1) (2n + 1)/6. For example. 1 + 2 + 3 = 6= 4-3 / 2. Works! What about l3 + 23 + 33 + ... + n3 =? Let's see whether we can make an educated guess. Expanding the right sides of our first two formulas gives So, the formula for the first sum is of the form an2 + bn + c and the formula for the second sum is of the form an3 + bn2 + cn + d (with just the right choices of coefficients). So, maybe, the next formula is of the form 13 + 23 + 33 + ... + n3 = an4 + bn3 + cn2 + dn + e. Here then is your mission: By assuming that this can really be done, find coefficients a, b, c, d, e that work. To answer this question you may want to use the Linear Algebra Toolkit (available online - google it) or other software. Just make sure that you record the input and output and justify what you are doing. (Hint: Substitute a few different values for n in this would-be formula to generate some linear equations in the unknowns a.b,c,d,e... Also, to make it a bit easier for yourself, note that the first two formulas even work for n = 0, if you interpret the sums on the left to add up to 0. So this should also work here.) (Bonus: 10 points) Prove that your formula really works. (Hint: One standard way of proving formulas like this is called Proof by Induction, You can learn how to do such a proof here http://nrich.maths.org/4718, but we'll also give a mini-introduction to this in class).
You wonder why they have slowed down so much, so you keep checking in every hour.
Time (hours)
1
2
3
4
5
6
Height (feet, inches)
3' 11"
4' 11"
5' 8"
6' 3"
6' 6"
6' 11"
After six hours you are worried the entire festival might fail, but you won't give up! You keep spreadsheeting.
Time (hours)
7
8
9
10
11
12
Height (feet, inches)
7' 4"
7' 8"
7' 11"
8' 3"
8' 6"
8' 9"
After 12 hours, you take a quick trip home to change clothes, and return to your spreadsheeting.
Time (hours)
13
14
15
16
17
18
Height (feet, inches)
8' 11"
9' 2"
9' 5"
9' 7"
9' 9"
10' 0"
Finally, late in the night, around 3am, it is complete. Your masterpiece. The Grand Cone-yan. But why did it take so long? It took a lot longer than 10 minutes. Why did the workers slow down so much? The first minute finished the first foot, but three feet took 30 minutes instead of 3 minutes. And now it's 3am!
The workers say that they filled at a steady rate of 13 gallons per hour, and that any faster would run the risk of drain freeze.
Your responses
1. What is wrong with the following argument?
If you really did 13 gallons per hour, then the 10 foot cone should have only taken 10 / 13 = 0.7 hours, less than an hour tops.
2. What is wrong the following question?
How do you convert gallons into feet?
3. Find an authoritative source for the volume of a cone. Give the formula and explain what each part of the formula means in this problem.
4. Most formulas will have both radius and height. Find a reasonable source for the relationship between the radius and height of an ice cream cone. Give an expression like
R = H
5. Calculate the volume of the cone using your formula. At a constant rate of 13 gallons per hour, how long should it have taken to fill up?
You may use the approximate formula V=0.02908882 h3 if the formula you discovered does not match at all.
6. There should still be a discrepancy between the V=0.02908882 h3 and the spreadsheet. Can you explain what likely happened?
