Tree Growth An evergreen nursery usually sells a certain type of shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by , where t is the time in years and h is the height in centimeters. The seedlings are 12 centimeters tall when planted .
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?
