Question

Apply Concepts A classmate converts 68 km to meters and gets 0.068 m as the answer. Explain why this answer is incorrect, and identify the likely source of the error

Short answer

The classmate's answer of 0.068 m is incorrect because they divided 68 km by 1000 instead of multiplying it by 1000. The correct conversion method is to multiply the number of kilometers by 1000, giving the correct answer of 68000 m. The error likely occurred due to not understanding the correct conversion between kilometers and meters.

Step-by-step solution

Understand the concept of conversion between kilometers and meters

To convert kilometers to meters, we should understand that 1 kilometer is equal to 1000 meters. Therefore, to convert any number of kilometers to meters, we need to multiply the number of kilometers by 1000.

Convert 68 km to meters correctly

Using the conversion concept from Step 1, we can convert 68 km to meters like this: \(68 \ \text{km} \times 1000 \ \frac{\text{m}}{\text{km}} = 68000 \ \text{m} \)

Identify the incorrect answer and error

The classmate gets an answer of 0.068 m for the conversion, which is incorrect. The correct answer is 68000 m, as we found in Step 2. The likely source of error in the classmate's calculation is that they divided 68 by 1000 instead of multiplying it by 1000. This led to the wrong conversion of 68 km to 0.068 m.

Explain why the answer is incorrect and the source of the error

The classmate's answer of 0.068 m is incorrect because they have not followed the correct conversion method between kilometers and meters. Instead of multiplying 68 km by 1000, they have divided it by 1000. The correct conversion method states that we have to multiply the number of kilometers by 1000 to convert them to meters, which leads to the correct answer of 68000 m.

Key concepts

Kilometers to Meters Conversion

Understanding how to convert from kilometers to meters is a fundamental aspect of unit conversion in chemistry and physics. Before delving into the equation, let's highlight the core principle: 1 kilometer is equivalent to 1,000 meters. This is a fixed conversion factor, meaning that no matter the number of kilometers, you will always multiply by 1,000 to convert to meters.

To illustrate, converting 68 kilometers to meters involves the following calculation:
\(68 \text{ km} \times 1000 \frac{\text{m}}{\text{km}} = 68000 \text{ m}\).

The mistake made by the classmate in the example was not just a simple oversight but indicates a fundamental misunderstanding of the conversion process. By dividing instead of multiplying, the classmate arrived at an answer that was significantly smaller than the correct value.

Dimensional Analysis

Dimensional analysis is not just a method; it's a critical thinking framework used in science and engineering to convert one unit of measure to another. It ensures that the conversion of units is done consistently and accurately. The process involves multiplying by a conversion factor that is a ratio of equivalent values, but with different units.

Here's how you approach dimensional analysis in the context of the example:

Conversion Factor:

1 kilometer is the same as 1,000 meters, so our conversion factor is \(1000 \frac{\text{m}}{\text{km}}\).

The Process:

To convert 68 km to meters, you line up the units so that kilometers will cancel out, leaving only the desired unit, meters: \(68 \text{ km} \times 1000 \frac{\text{m}}{\text{km}}\). After canceling out 'km,' you get the product in meters.

Using dimensional analysis systematically not only helps in performing conversions correctly but also in checking the reasonableness of your answers. If your unit of measure becomes smaller when it should become larger (as was the case when kilometers were incorrectly converted to a smaller number of meters), it can be a red flag that the conversion has not been done properly.

Scientific Notation Error

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It's often used in science and engineering to simplify numbers and calculations. However, errors can occur if the notation is not used properly. In our example, the classmate likely made an error related to the understanding of scientific notation.

The number 68 can be written in scientific notation as \(6.8 \times 10^1\), whereas the incorrect result of 0.068 m can be written as \(6.8 \times 10^{-2}\) m. The powers of 10 in scientific notation are crucial — they determine the actual value of the number. Here, the error was in using a negative exponent when a positive exponent was necessary.

Correctly applying scientific notation would reinforce that when converting to a larger unit (kilometers to meters), the exponent will increase, indicating a larger number. For example, converting 0.068 kilometers to meters should have involved shifting the decimal to the right, which in scientific notation would result in an increase of the exponent in powers of ten, correctly giving us \(6.8 \times 10^4\) meters, or 68,000 meters. This type of awareness can help prevent similar errors in the future.