You are handed a 10.0 cm stack of new one dollar bills. Assume the thickness of a dollar bill is 1.6 times thicker than your text book paper (textbook paper = 63 um) Estimate the number of dollars in your stack.
The correct answer and explanation is :
To estimate the number of one-dollar bills in a 10.0 cm stack, we need to use the thickness of a dollar bill, which is given in terms of the thickness of textbook paper.
Step 1: Calculate the Thickness of a Dollar Bill
We are told that the thickness of a dollar bill is 1.6 times thicker than the thickness of textbook paper. The thickness of a sheet of textbook paper is 63 micrometers (µm).
- Thickness of a dollar bill:
[
\text{Thickness of one bill} = 1.6 \times 63 \, \mu m = 100.8 \, \mu m
]
We now know that the thickness of one dollar bill is 100.8 µm.
Step 2: Convert the Stack Height to Micrometers
The stack is given as 10.0 cm. To work with compatible units, we first convert the height of the stack from centimeters to micrometers (since the thickness of a dollar bill is in micrometers):
- 1 cm = 10,000 µm, so:
[
10.0 \, \text{cm} = 10.0 \times 10,000 \, \mu m = 100,000 \, \mu m
]
Step 3: Calculate the Number of Dollar Bills
To estimate the number of dollar bills in the 10.0 cm stack, we divide the total height of the stack by the thickness of one dollar bill:
[
\text{Number of bills} = \frac{\text{Height of the stack}}{\text{Thickness of one bill}} = \frac{100,000 \, \mu m}{100.8 \, \mu m} \approx 991.2
]
Since the number of bills must be a whole number, we round this to approximately 991 bills.
Final Answer:
The stack of 10.0 cm contains about 991 one-dollar bills.
Explanation
The key to solving this problem is using unit conversion and applying the given relationships. The thickness of the stack is divided by the thickness of a single dollar bill to estimate the number of bills. The solution incorporates basic mathematical operations like multiplication and division, as well as unit conversion between centimeters and micrometers to ensure consistent units throughout the calculation. While the estimate provides a close approximation, small variations in paper thickness or measurement methods could yield a slightly different result in a real-world scenario.