---
title: Pumpkin Coins Estimate
tags:
  - topic/estimating
  - type/idea
up: "[[pdi-estimating]]"
---
# 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

$$
\begin{align*}
M_{B} &= V_{B} \times D
\end{align*}
$$