Question

Some thieves, escaping after a bank robbery, drop a sack of money on the sidewalk. (a) Estimate the mass of the sack if it contains \(\$ 5000\) in half- dollar coins. (b) Estimate the mass if the sack contains \(\$ 1000000\) in \(\$ 20\) bills.

Short answer

The sack with coins weighs 113.4 kg and the sack with bills weighs 50 kg.

Step-by-step solution

Understanding the Coin Mass Problem

We need to find the mass of a sack containing $5000 in half-dollar coins. Each half-dollar coin is worth $0.50. First, determine how many coins make up $5000. This is done by dividing $5000 by $0.50 per coin.

Calculate Number of Coins

Calculate the number of half-dollar coins: \[ \text{Number of coins} = \frac{5000}{0.50} = 10000 \text{ coins} \]

Mass per Half-Dollar Coin

Next, we need to know the mass of a half-dollar coin. A typical half-dollar coin (e.g., Kennedy) weighs about 11.34 grams.

Find Total Mass of Coins

Multiply the number of coins by the mass of each coin to find total mass: \[ \text{Total mass} = 10000 \times 11.34 \text{ g} = 113400 \text{ g} \] Convert grams to kilograms (1 kg = 1000 g): \[ 113400 \text{ g} = 113.4 \text{ kg} \]

Understanding the Bill Mass Problem

Now, estimate the mass of a sack containing $1000000 in $20 bills. First, determine how many $20 bills are needed to make $1000000. This is done by dividing $1000000 by $20 per bill.

Calculate Number of Bills

Calculate the number of $20 bills: \[ \text{Number of bills} = \frac{1000000}{20} = 50000 \text{ bills} \]

Mass per $20 Bill

Next, find the mass of a $20 bill. A typical U.S. bill weighs about 1 gram.

Find Total Mass of Bills

Multiply the number of $20 bills by the mass of each bill to find total mass: \[ \text{Total mass} = 50000 \times 1 \text{ g} = 50000 \text{ g} \] Convert grams to kilograms: \[ 50000 \text{ g} = 50 \text{ kg} \]

Key concepts

Unit Conversion

Unit conversion is a fundamental skill in physics and many other scientific disciplines. It allows us to appropriately change units from one type to another, such as converting grams to kilograms or miles to kilometers.
To perform unit conversion effectively, follow these basic steps:

• Identify the units you are converting from and to.
• Use a conversion factor, which is a number that will change the original quantity into a new unit.
• Multiply the original quantity by the conversion factor.

For example, in the exercise at hand, we initially calculate the mass in grams but need the result in kilograms. To convert grams to kilograms, use the conversion factor: 1 kg = 1000 g. This means to find the mass in kilograms, you divide the number of grams by 1000. That's why 113400 g becomes 113.4 kg. This simple method keeps calculations accurate and ensures you're using the correct unit for your context.

Weight of Currency

Understanding the weight of money might seem trivial, but it has practical applications, especially in scenarios where we deal with large sums of physical currency, like in this exercise. Each type of currency has a specific weight.
For coins, the weight depends on the material and size of the coin. For instance, a half-dollar coin, as noted, weighs about 11.34 grams. This fixed mass allows us to calculate the total weight when we have a specific number of these coins. Paper bills, like the U.S. $20 bill, are quite different. Each bill weighs approximately 1 gram, making calculations straightforward. When estimating the mass of money, simply multiply the number of currency units by the weight of one. These calculations help accurately determine the physical heft of money, which can be crucial in logistics, security, and planning for physical movement.

Practical Estimation

Practical estimation skills involve making educated guesses or approximate calculations to gain a basic understanding of a situation without needing exact measures.
Using practical estimation, as in the exercise, helps in simplifying assumptions and applying straightforward calculations to obtain rough yet reasonable solutions. Here's how practical estimation works:

• Break down the problem into manageable parts, such as estimating the number of coins or bills, as seen in the exercise.
• Use rough calculations to arrive at a viable solution, like estimating the total weight by multiplying the number of items by their individual weight.
• Remember that these are approximations and meant to get you close to the answer rather than provide exact figures.

This skill is essential in real-world scenarios where precise data may not be readily available, allowing for action-based decisions in the planning and execution of tasks.