Problem 64 - Chemistry: The Central Science

Question

The US 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) At the beginning of 2007 , the national debt was \(\$ 8.7\) trillion. How many stacks like the one described would be necessary to pay off this debt?

Step-by-Step Solution

Answer

To reach the height of the Washington Monument (575 ft), 113,043 quarters would be needed. The stack would weigh 1,413.6 lbs and contain $28,260.75. Approximately 307,700,000 such stacks would be needed to pay off the \$8.7 trillion national debt.

Step 1: Convert feet to millimeters

To compare the height of the Washington Monument with the thickness of a quarter, we must first convert its height from feet to millimeters: \(1 \mathrm{~ft} = 304.8 \mathrm{~mm}\) Now multiply the given height (575 ft) by the conversion factor: \(575 \mathrm{~ft} \times 304.8 \frac{\mathrm{mm}}{\mathrm{ft}} = 175260 \mathrm{~mm}\)

Step 2: Calculate the number of quarters

To find the number of quarters needed to reach this height, we will divide the height in millimeters (175260 mm) by the thickness of one quarter (1.55 mm): \(\frac{175260 \mathrm{~mm}}{1.55 \mathrm{~mm/quarter}} = 113042.6 \mathrm{~quarters}\) Since we can't have a fraction of a quarter, we will round up to 113043 quarters. #b) Finding the total weight of the stack.#

Step 3: Calculate the total weight

Multiply the number of quarters (113043) by the mass of one quarter (5.67 g): \(113043 \mathrm{~quarters} \times 5.67 \mathrm{~g/quarter} = 640,753.81 \mathrm{~g}\)

Step 4: Convert grams to pounds

To convert the total weight to pounds: \(1 \mathrm{~lb} = 453.592 \mathrm{~g}\) Now, divide the total weight in grams (640,753.81) by the conversion factor: \(\frac{640,753.81 \mathrm{~g}}{453.592 \frac{\mathrm{g}}{\mathrm{lb}}} = 1413.6 \mathrm{~lb}\) #c) Finding the total value of the stack.#

Step 5: Calculate the total value

Each US quarter is worth 25 cents or 0.25 dollars. Multiply the number of quarters (113043) by the value of one quarter (0.25 dollars): \(113043 \mathrm{~quarters} \times 0.25 \mathrm{\$/quarter} = \$28,260.75\) #d) Finding the number of stacks needed to pay off the national debt.#

Step 6: Convert trillions to dollars

The national debt is 8.7 trillion dollars. We have to convert trillions to dollars: \[8.7 \times 10^{12} \mathrm{\$}\]

Step 7: Calculate the number of stacks

Divide the national debt (8.7 x 10^12 dollars) by the value of one stack (28260.75 dollars): \(\frac{8.7 \times 10^{12} \mathrm{\$}}{28,260.75 \mathrm{\$/stack}} = 3.077 \times 10^8 \mathrm{~stacks}\) Since we can't have a fraction of a stack, we will round up to 307,700,000 stacks.

Key Concepts

Mass and Weight in Chemistry

Understanding mass and weight is crucial in chemistry, especially in converting units like grams to pounds. Mass refers to the amount of matter in an object and is measured in grams in the metric system. Meanwhile, weight is the force exerted by gravity on that mass. This is generally measured in pounds in the imperial system.

When dealing with calculations such as finding the weight of a stack of quarters, we begin by identifying the mass of a single quarter, here given as 5.67 grams. To determine the total weight of many such quarters, we multiply the mass of one by the total number of quarters. In this exercise, we get the total mass as 640,753.81 grams. To convert this mass into weight in pounds, we use the conversion factor where 1 pound equals 453.592 grams.

Using these principles, you can convert the total mass from grams to pounds by dividing:

• Total weight (pounds) = Total mass (grams) / 453.592

Such conversions are common in chemistry when switching between unit systems.

Stacking Calculations

In exercises that involve stacking, such as our quarters, it's all about calculating height and ensuring proper unit conversion. Here, we are asked to stack quarters to match the height of the Washington Monument at 575 feet. To find out how many quarters are needed, we first convert this height to millimeters (since quarter thickness is given in millimeters).

This involves using the conversion factor of 1 foot being equivalent to 304.8 millimeters. Hence, converting 575 feet, we find the height in millimeters is 175,260 mm.

Since each quarter is 1.55 mm thick, to find how many quarters are needed, divide the total millimeter height by the millimeter thickness of one quarter:

• Number of quarters = Height in mm / Thickness of one quarter (mm)

This gives a result of 113,042.57 quarters, rounded up to 113,043 quarters to ensure the full height is reached.

Financial Estimation Using Chemistry

Financial estimations in chemistry involve determining values or costs related to chemical processes or experiments. In this exercise, the value of a stack of quarters is calculated. Each quarter is worth $0.25.

To find the total value of the stack, multiply the number of quarters by their individual value. So, for 113,043 quarters:

• Total value = Number of quarters × Value per quarter

This results in $28,260.75.

Such estimations can also be stretched to large scales, like national debt proportions, by comparing the total stack value to larger sums. With a debt of $8.7 trillion, we calculate how many such stacks would be needed:

• Number of stacks = Total debt / Value per stack

This reveals that approximately 307,700,000 stacks are necessary to match the debt in 2007.