Problem 48 - Algebra 1

Question

At the 1998 winter Olympics, Marianne Timmer of the Netherlands won the women's 1000 -meter speed skating race with a time of 76.51 seconds. Use a calculator to find Timmer's rates in parts (a) - (d) following each method below. Do the two methods give the same= final result after rounding to the nearest tenth in part (d)? Method \(\mathbf{1}\) Round any decimal answers to the nearest tenth before going on. Method 2 Use the full calculator display until you round for the final answer. a. speed in meters per second b. speed in meters per minute c. speed in meters per hour d. speed in kilometers per hour

Step-by-Step Solution

Answer

a. Speed in meters/second: 13.1 (Method 1), 13.07029475 (Method 2)\nb. Speed in meters/minute: 786.0 (Method 1), 784.217685 (Method 2)\nc. Speed in meters/hour: 47160.0 (Method 1), 47053.0611(Method 2)\nd. Speed in km/hour: 47.2 (Method 1), 47.053 (Method 2)\nAfter rounding to the nearest tenth, the results with both methods are not the same.

Step 1: Speed in meters per second

Speed is calculated by dividing distance by time. Therefore, the speed in meters per second can be calculated as \( \frac{1000 meters}{76.51 seconds}\)

Step 2: Speed in meters per minute

To convert speed in meters per second into meters per minute, multiply the obtained speed by 60 (as there are 60 seconds in a minute).

Step 3: Speed in meters per hour

To convert speed in meters per minute to meters per hour, multiply the speed by 60 (as there are 60 minutes in an hour).

Step 4: Speed in kilometers per hour

To convert speed in meters per hour to kilometers per hour, divide the speed by 1000 (as there are 1000 meters in a kilometer).

Step 5: Comparing the results

In order to answer the second part of the problem, the final results obtained using both methods should be compared. The results would be considered the same if, after rounding to the nearest tenth, they are equal.

Key Concepts

Understanding Rate of Change

In the context of speed, the rate of change refers to how quickly a particular distance is covered over time. It's a measure of velocity and is commonly expressed in units such as meters per second (m/s), meters per minute (m/min), or kilometers per hour (km/h). When calculating the rate of change for Olympic speed skater Marianne Timmer, the process involves dividing the distance she skated, 1000 meters, by the time it took her, which was 76.51 seconds. This gives us her speed in meters per second.

The rate of change is not only about speed in physics; it's a versatile concept that can apply to various forms of change over time, such as population growth, temperature fluctuations, or the stock market. Understanding this concept provides a foundational basis for solving problems related to motion and dynamics.

Unit Conversion Simplified

Unit conversion is a fundamental skill in various fields such as science, engineering, and everyday life. It allows you to translate a quantity expressed in one set of units to another. In our exercise, we converted Marianne Timmer's skating speed from meters per second to meters per minute, then to meters per hour, and finally to kilometers per hour. We multiplied and divided by conversion factors that represent the equivalent amounts in different units. For instance, to get from meters per second to meters per minute, we used the fact that there are 60 seconds in a minute and therefore multiplied by 60.

Understanding unit conversions ensures accurate comparisons and calculations, especially when dealing with international datasets that may use varied measurement systems. It's always crucial to keep track of the units you're working with to avoid errors in your solutions.

Decimal Rounding Methods

Decimal rounding methods are mathematical procedures used to reduce the number of digits right of the decimal point. In our exercise, two different rounding methods were used: one rounds to the nearest tenth before continuing calculations, while the other uses the full calculator display and rounds only at the final step.

When to Round

Deciding when to perform rounding in a multi-step calculation can influence your final result. Rounding too soon may cause a 'loss of precision,' which refers to potentially significant deviations from the actual value if the intermediary values are not precise.

Rules of Rounding

Common rules to follow while rounding decimals include rounding up if the next digit is 5 or above, and down if it's 4 or below. However, depending on the level of precision required, there are other methods like truncation, rounding to even, and more.

Rounding is not just a numerical trick; it is a tool that helps in reducing complexity and making numbers manageable, especially when exact values are not necessary for practical purposes.