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?

Short answer

(a) 113,039 quarters are required to reach the height of the Washington Monument. (b) The weight of the stack would be 640.83 kg. (c) The total amount of money in the stack would be $28,259.75. (d) Approximately 573,492,330 stacks would be needed to pay off the U.S. National Debt as of October 28, 2012.

Step-by-step solution

Convert the height to millimeters

The height of the Washington Monument is given in feet (575 ft), we will convert it to millimeters to match the thickness's unit. 1 ft = 30.48 cm = 304.8 mm So, the height is 575 ft × 304.8 mm/ft = 175,260 mm.

Calculate the number of quarters required to reach this height

Given the thickness of one quarter is 1.55 mm, we can calculate the number of quarters required to reach 175,260 mm. Number of quarters = Height / Thickness = \( \frac{175{,}260}{1.55} \) = 113,039.

Calculate the weight of the stack

With the total number of quarters (113,039) and the mass of one quarter (5.67g), we can find the total mass (weight) of the stack. Total Weight = Number of quarters × Mass of one quarter = 113,039 × 5.67g = 640,831.13g.

Convert the weight to kilograms

To express the weight in a more common unit, convert grams to kilograms. 1 kg = 1000g Weight of the stack = 640,831.13g × 1 kg/1000g = 640.83 kg

Calculate the total amount of money in the stack

Since each quarter is worth 0.25 USD, we can find the total amount of money in the stack by multiplying the number of quarters by the value of one quarter. Total Amount = 113,039 × 0.25 USD = $28,259.75.

Calculate the number of stacks needed to pay off the National Debt

Given the National Debt is $16,213,166,914,811, we need to find how many stacks of the quarters described earlier would be required to pay off this debt. Number of stacks = National Debt / Amount in one stack = \( \frac{16{,}213{,}166{,}914{,}811}{28{,}259.75} \) ≈ 573,492,330. The exercise is solved, and the answers are as follows: a) 113,039 quarters are needed to reach the height of the Washington Monument. b) The stack of quarters would weigh 640.83 kg. c) The total amount contained in that stack would be $28,259.75. d) About 573,492,330 such stacks would be necessary to pay off the National Debt as of October 28, 2012.

Key concepts

Weight Calculation

Understanding weight calculation is a fundamental skill that helps in various aspects of science and everyday life. In this exercise, we looked at the total weight of a stack of quarters.

To calculate the weight, you first need to know the mass of a single item—in this case, a quarter, which is 5.67 grams. Once you have the mass, and you know how many items there are, you simply multiply these two numbers to get the total weight.

• Weight of 1 quarter = 5.67 grams
• Total quarters = 113,039
• Total weight = 5.67 grams × 113,039 quarters

This resulted in a total weight of 640,831.13 grams. To present this in a more standard unit, you convert grams to kilograms by dividing by 1000 (since 1 kilogram equals 1000 grams). This conversion simplifies and clarifies the information, allowing us to express the weight as 640.83 kg. In summary, the steps involved in calculating weight are straightforward multiplication and simplification through unit conversion.

Height Conversion

Height conversion is crucial when different units of measure are needed for comparison or calculation. The exercise required converting the height of a monument in feet to millimeters, aligning with the metric unit used for thickness.

To initiate a proper conversion, you need a conversion factor, which in this case, is that 1 foot equals 304.8 millimeters. Therefore, converting feet to millimeters is done by multiplying the number of feet by 304.8.

• Given Height of 575 ft
• Conversion factor = 304.8 mm/ft
• Converted height = 575 ft × 304.8 mm/ft

Executing this multiplication yields 175,260 millimeters, which seamlessly aligns with the thickness measurement in the metric system. Such conversions ensure a common frame of reference when comparing different values or when involved in multiple calculations, aiding clarity and precision.

Monetary Value Calculation

Monetary value calculation helps to determine the actual worth of items when given quantity and single item value, which is essential in financial contexts.

Every quarter is equivalent to 0.25 USD. Knowing this, the total monetary value can be calculated by multiplying the number of quarters by the value of each.

• Value of 1 quarter = 0.25 USD
• Total number of quarters = 113,039
• Total monetary value = 113,039 quarters × 0.25 USD/quarter

This computation results in an accumulated value of $28,259.75 for the entire stack of quarters. Understanding such calculations is not only helpful for solving mathematical problems but also reflects practical daily activities, such as budgeting and making financial decisions.