Problem 75 - Chemistry The Central Science

Question

The U.S. quarter has a mass of 5.67 \(\mathrm{g}\) and is approximately 1.55 \(\mathrm{mm}\) thick. (a) How many quarters would have to be stacked to reach 575 \(\mathrm{ft}\) , the height of the Washington Monument? (b) How much would this stack weigh? (c) How much money would this stack contain? (d) The U.S. National Debt Clock showed the outstanding public debt to be \(\$ 16,213,166,914,811\) on October \(28,2012 .\) How many stacks like the one described would be necessary to pay off this debt?

Step-by-Step Solution

Answer

In summary, \(113,039\) quarters are needed to reach the height of the Washington Monument, with the stack weighing approximately \(640.81 kg\) and containing $28,259.75. To pay off the U.S. national debt at that time, approximately \(573,448,261\) stacks of quarters would be necessary.

Step 1: (a) Finding the number of quarters

First, we need to convert the height of the Washington Monument from feet to millimeters. There are 1 foot = 304.8 millimeters, so: \(575 ft * 304.8 \frac{mm}{ft} = 175260 mm\) Now, we will divide this height by the thickness of a U.S. quarter (1.55mm) to find the number of quarters needed: \(\frac{175260 mm}{1.55 \frac{mm}{quarter}} = 113039 \text{ quarters}\) So, approximately 113,039 quarters would need to be stacked to reach the height of the Washington Monument.

Step 2: (b) Finding the weight of the stack

We have to multiply the mass of a single quarter (5.67g) by the number of quarters in the stack (113,039) to find the total weight of the stack: \(5.67 \frac{g}{quarter} * 113039 \text{ quarters} = 640810.13g\) To make it more understandable, we can convert the weight to kilograms by dividing by 1000: \(\frac{640810.13g}{1000} = 640.81 kg\) So, the stack of quarters would weigh approximately 640.81 kilograms.

Step 3: (c) Finding the total amount of money in the stack

To find the total amount of money in the stack of quarters, we need to multiply the number of quarters (113,039) by the value of a single U.S. quarter (0.25 dollars or 25 cents): \(113039 \text{ quarters} * 0.25 \frac{dollars}{quarter} = 28259.75 dollars\) So, the stack of quarters would contain $28,259.75.

Step 4: (d) Finding the number of stacks needed to pay off the U.S. national debt

To find the number of stacks needed to pay off the U.S. national debt, we would divide the national debt amount (\(16,213,166,914,811) by the total amount of money in a single stack of quarters (\)28,259.75): \(\frac{16213166914811 dollars}{28259.75 \frac{dollars}{stack}} = 573448260.23 \text{ stacks}\) Since we cannot have a partial stack, we will round up to the nearest whole number: \(573,448,261 \text{ stacks}\) Therefore, approximately 573,448,261 stacks of quarters would be necessary to pay off the U.S. national debt at that given time.

Key Concepts

Metric System

The metric system is a simple and cohesive measuring system, ideal for scientific calculations. It uses units like meters for length, grams for mass, and liters for volume. These units are inter-convertible using multiples of ten, making the system convenient and accessible.
In the problem above, converting the height of the Washington Monument from feet into millimeters required an understanding of the metric system's conversion factor: 1 foot is equivalent to 304.8 millimeters.

• To convert from feet to millimeters, multiply the number of feet by 304.8.
• Metric calculations ensure that units are consistent, reducing errors in computation.

In practice, many countries use the metric system, thus increasing accuracy in global communication and exchange. Understanding this system not only aids in local calculations as shown in the exercise but also prepares you to handle international metrics effectively.

Currency Calculations

Currency calculations require basic multiplication to determine values in different contexts. In this exercise, the task was to define how much a specific number of quarters is worth. Each U.S. quarter is valued at 0.25 dollars.
This concept helps:

• Understand the numerical equivalence of different coins and bills in relation to the dollar.
• Determine total monetary values by multiplying individual coin values by their respective counts.

The same principle scales from small sums in your own pocket to large sums, such as calculating how many stacks of quarters would be needed to pay off a national debt.
Currency conversions follow straightforward arithmetic operations, making them intuitive for budgeting and financial analysis.

Volume and Mass

Volume and mass are measures of the amount of space an object occupies and the amount of matter it contains, respectively. In this exercise, the mass of each quarter was crucial to computing the total weight of the stack.
Understanding mass:

• The mass of a single quarter is 5.67 grams.
• Multiplying this value by the number of quarters gives the total mass of the stack.

Mass is typically reported in grams or kilograms in the metric system.
Understanding volume, while less directly relevant to this exercise, helps when considering how much physical space an object or stack occupies. In various scenarios, recognizing both volume and mass can lead to informed decisions; for example, shipping costs, material usage, and storage concerns might all consider these properties.