vault backup: 2025-10-27 16:34:32

pumpkin-coins-estimate.md

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