Question

What is the area in square millimeters of a rectangle whose sides are \(2.44 \mathrm{~cm} \times 6.077 \mathrm{~cm}\) ? Express the answer to the proper number of significant figures.

Short answer

The area is \( 1.48 \times 10^3 \, \text{mm}^2 \).

Step-by-step solution

Convert centimeters to millimeters

Since the problem asks for the area in square millimeters, we first convert the dimensions from centimeters to millimeters. Recall that 1 cm equals 10 mm. Therefore: - The shorter side: 2.44 cm = 24.4 mm - The longer side: 6.077 cm = 60.77 mm.

Calculate the area in square millimeters

To find the area, multiply the length by the width in millimeters:Area \(= 24.4 \, \text{mm} \times 60.77 \, \text{mm} = 1483.788 \, \text{mm}^2\).

Apply significant figures

The number of significant figures is determined by the measurement with the fewest significant figures. - 2.44 cm has three significant figures, while 6.077 cm has four.Thus, the area should be expressed with three significant figures:Area \(= 1.48 \times 10^3 \, \text{mm}^2\).

Key concepts

Unit Conversion

Unit conversion is the process of changing one unit of measurement to another, ensuring that the comparison between quantities remains consistent. In mathematical problems and scientific calculations, this skill is especially important when working with different units of measurement. In this specific exercise, we need to convert the dimensions of the rectangle from centimeters to millimeters, as the final area needs to be in square millimeters.

The basic conversion factor to remember here is:

• 1 centimeter (cm) = 10 millimeters (mm).

To convert centimeters to millimeters, you simply multiply the centimeter measurement by this conversion factor. For example,

• 2.44 cm becomes 24.4 mm.
• 6.077 cm becomes 60.77 mm.

By converting each dimension, you can now calculate areas or perform further operations using the same units, which is crucial for accuracy in mathematical computations.

Area Calculation

Calculating the area of a rectangle involves multiplying its length by its width. When the dimensions are given in the same unit, the calculation becomes straightforward. For the rectangle in this exercise, after converting the sides to millimeters, we multiply:

• Length: 24.4 mm
• Width: 60.77 mm

The formula for the area of a rectangle is:\[\text{Area} = \text{length} \times \text{width}\]Using our numbers, the calculation is \[24.4 \, \text{mm} \times 60.77 \, \text{mm} = 1483.788 \, \text{mm}^2\]
It’s important to keep track of units and ensure that all are consistent to get an accurate area calculation in the desired units. This area represents the total surface that lies within the boundaries of the rectangle.

Rectangles

Rectangles are four-sided polygons with opposite sides that are equal and parallel. In a rectangle, all internal angles are right angles, which means each measures 90 degrees. This unique property significantly simplifies area calculations.

When identifying or calculating properties of a rectangle, keep in mind:

• The opposite sides are equal in length.
• All the angles are 90 degrees.
• The formula for area is always the product of its length and width.

In our problem, it's crucial to ensure that the sides provided are treated as length and width rather than hypothesizing about more complex geometric properties found in other shapes, like squares or rhombuses. Focusing on these straightforward characteristics helps in correctly understanding and applying formulae related to rectangles.