Problem 49 - College Physics

Question

The top of a rectangular table measures \(1.245 \mathrm{~m}\) by \(0.760 \mathrm{~m} .\) (a) The smallest division on the scale of the measurement instrument is \((1) \mathrm{m},(2) \mathrm{cm},(3) \mathrm{mm} .\) Why? (b) What is the area of the tabletop?

Step-by-Step Solution

Answer

(a) Smallest division is in millimeters; (b) Area is 0.9462 m².

Step 1: Understanding Instrument Measurement Units

To determine the smallest division of the measurement instrument, examine the precision in the given measurements: 1.245 m and 0.760 m. It is measured to the third decimal place. For metric measurements, 1 m = 100 cm = 1000 mm. So, if the smallest division is 1 m, the precision would be whole meters (no decimal); if it's 1 cm, the precision would be two decimals (0.01 m); and if it's 1 mm, the precision would be three decimals (0.001 m). The precision of three decimals matches the millimeter measurement.

Step 2: Calculating the Area of the Table

The area of a rectangle is calculated using the formula: \( ext{Area} = ext{length} imes ext{width} \). Plug in the given measurements for the table, \( ext{Area} = 1.245 \) \( ext{m} \times 0.760 \) \( ext{m} \).

Step 3: Solving the Area Calculation

Perform the multiplication: \( 1.245 \times 0.760 = 0.9462 \). This result gives us the area of the table in square meters.

Key Concepts

Measurement Precision

When it comes to measuring any physical quantity, being precise is crucial. Precision refers to how detailed and exact a measurement is. In our rectangular table exercise, we have dimensions given as 1.245 meters and 0.760 meters. The smallest measurable unit of the instrument affects the measurement precision. Here's how it works:

• If the smallest division on the instrument is 1 meter, the measurement would appear as a whole number, like just 1 meter, with no decimal places. However, in our case, we see measurements up to the third decimal place.
• If it's 1 centimeter, equivalent to 0.01 meters, you would expect precision up to two decimal places, such as 1.24 m.
• Given the measurements provided, which go up to three decimal places (e.g., 1.245 m), the smallest division must be 1 millimeter, which equals 0.001 meters.

This detailed level helps ensure accuracy and helps us use measurements confidently across various calculations.

Rectangular Area Calculation

The area of a rectangle is one of the fundamental concepts in geometry. It tells you how much surface a shape covers. To calculate the area of a rectangle, you need to multiply its length by its width, both of which are conveniently given in our problem.In the table's case, it measures 1.245 meters by 0.760 meters. Using the formula for area:\[ \text{Area} = \text{length} \times \text{width} \]Simply plug in the numbers: \[ \text{Area} = 1.245 \times 0.760 = 0.9462 \text{ m}^2 \]This formula works across many scenarios, whether you want to fit a rug to your floor or paint a wall. Always ensure your units for length and width match (like both in meters) to avoid any unit conversion errors.

Metric Units

The metric system is a comprehensive and widely used international standard of measuring. It is essential to grasp metric units, especially in science contexts, to easily handle calculations effectively. Here are some key points about metric units:

• A meter (m) is the base unit of length in the metric system. It's particularly convenient for measuring larger dimensions such as room sizes or distances.
• A centimeter (cm) is one hundredth of a meter. It's commonly used for smaller items like books or furniture dimensions.
• Lastly, a millimeter (mm) is a thousandth of a meter. It's ideal for even smaller objects, such as screws or beads, where high precision is required.

These units connect like gears in a clock, making conversions straightforward: 100 centimeters in a meter and 1000 millimeters in a meter. If you are familiar with the metric system, converting between these units becomes second nature, significantly aiding in any scientific calculation or everyday measurement task.