76 lines
1.8 KiB
Markdown
76 lines
1.8 KiB
Markdown
---
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tags:
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- topic/estimating
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- type/idea
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title: Pumpkin Coins Estimate
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---
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# Pumpkin Coins Estimate
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## Announcement
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> [!quote]
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> ...
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> There's a pumpkin filled with coins just waiting for someone to nail the number.
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> Think you've got what it takes to guess the total value inside?
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>
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> 💰 Guess the Dollar Value in the Pumpkin
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>
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> ...
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>
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> 🕒 All votes due by 3:00 PM on Wednesday, 10/29
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>
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> Bring your best guess---the closest estimate wins big!
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> ...
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## Strategy
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### Expected Coin
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Estimate expected value and volume for a randomly selected coin.
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| Coin | | Frequency $F$ | Monetary Value $M$ | Radius $r$ | Thickness $h$ | Volume $V$ |
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| ------- | --- | ------------- | ------------------:| ---------- | ------------- | ---------- |
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| Quarter | | | 0.25 | | | |
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| Dime | | | 0.10 | | | |
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| Nickel | | | 0.05 | | | |
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| Penny | | | 0.01 | | | |
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$$
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V = \pi r^2 \times h
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$$
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### Expected Monetary Density
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$$
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D = \frac{M_{E}}{V_{E}} \times E
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$$
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where
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* $M_{E}$ = Monetary value of the Expected Coin
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* $V_{E}$ = Cylindrical volume of the Expected Coin
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* $E$ = Packing efficiency
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> [!note] Estimating Packing Efficiency
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> Estimating $E$ will be tricky.
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### Volume of the "Bowl"
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Let the "bowl" be a hemispherical volume with radius $r_{B}$ enclosing all coins.
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The volume of the bowl $V_{B}$, the liquid volume occupied by the coins, is given by
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$$
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V_{B} = \frac{2}{3} \pi r_{B}^3
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$$
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### Total Monetary Value
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The total monetary value of all coins in the bowl is given by
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$$
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\begin{align*}
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M_{B} &= V_{B} \times D
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\end{align*}
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$$
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