Problem 17 - Chemistry

Question

(a) A 12 -inch pizza has an area of \(113.112\) square inches \(\left(\mathrm{in}^{2}\right)\). What is the area in units of \(\mathrm{cm}^{2}\) ? \((1\) inch \(=2.54 \mathrm{~cm}\), exactly \()\) (b) A can of soda has a volume of \(355 \mathrm{~mL}\). What is the volume expressed in SI units of \(\mathrm{m}^{3}\) ?

Step-by-Step Solution

Answer

(a) 729.04 cm²; (b) 0.000355 m³.

Step 1: Convert inches to centimeters

First, we need to convert the area of the pizza from square inches to square centimeters. Given that 1 inch is equal to 2.54 cm, we need to convert 12 inches to centimeters to understand the dimensions in centimeters. Therefore:\[\text{1 inch} = 2.54 \text{ cm, then, } (1 \text{ inch})^2 = (2.54 \text{ cm})^2\]This converts the square inches to square centimeters.

Step 2: Calculate area in square centimeters

Since 1 square inch is equal to \((2.54)^2\) square centimeters, we can multiply the area in square inches by this factor:\[113.112 \text{ in}^2 \times (2.54 \text{ cm/in})^2 = 113.112 \times 6.4516 \text{ cm}^2 \]Calculating the product gives us the area in square centimeters.

Step 3: Perform Conversion Calculation

Now, perform the calculation:\[113.112 \times 6.4516 = 729.04 \text{ cm}^2\]Hence, the area of the pizza in square centimeters is \(729.04 \text{ cm}^2\).

Step 4: Convert milliliters to cubic meters

Next, we convert the volume of the can of soda from milliliters to cubic meters. First, recognize that 1 milliliter (mL) equals \(1 \times 10^{-6}\) cubic meters (\(\text{m}^3\)). Therefore, calculate the volume by:\[355 \text{ mL} = 355 \times 10^{-6} \text{ m}^3\]

Step 5: Calculate volume in cubic meters

Perform the multiplication:\[355 \times 10^{-6} = 0.000355 \text{ m}^3\]Thus, the volume of the soda can in cubic meters is \(0.000355 \text{ m}^3\).

Key Concepts

Area Conversion

Converting units of area can seem tricky at first, but is very straightforward once you understand the process. Area conversion involves changing the units that measure the size of a two-dimensional space, like the surface of a pizza.

To convert from square inches to square centimeters, one must first know how inches relate to centimeters linearly. Since 1 inch equals 2.54 centimeters, to convert square inches to square centimeters, we must also square the conversion factor.

Thus, 1 square inch is \(2.54^2\) square centimeters, which equals approximately 6.4516 square centimeters.

This means to convert any number of square inches to square centimeters, you simply multiply by 6.4516. For example, if a pizza has an area of 113.112 square inches, it can also be expressed as 113.112 multiplied by 6.4516, resulting in about 729.04 square centimeters.

Volume Conversion

Volume conversion is similar to area conversion but involves three-dimensional space. If converting liquid volume from milliliters (mL) to cubic meters (m³), understanding the metric relationship is crucial.

The metric system often uses the liter as a standard, where 1 milliliter equals 1 cubic centimeter. More conveniently for mathematical transformations, 1 mL equals 1 x 10^{-6} cubic meters.

Thus, converting milliliters to cubic meters involves multiplying the volume in milliliters by 10^{-6}. For instance, a standard soda can holding 355 mL would convert to 0.000355 m³. This clear conversion shows how much space the soda occupies in cubic meters, offering an insight into three-dimensional volume measurement.

Metric System

The metric system is a globally recognized and used measurement system, known for its simplicity and ease of conversion.

It uses units like meters, liters, and grams, each having easily understandable fractional extensions based on powers of ten. For this reason, converting within the metric system is often more straightforward compared to other systems such as the imperial system.

Helpful aspects of the metric system include:

• Base ten simplification - easy multiply/divide by powers of ten.
• Uniformity - the same prefixes (kilo-, centi-, milli-) across different types of measurements.

These features make unit conversion in the metric system intuitive, which is particularly useful for students studying science and mathematics.

Measurement Units

Measurement units form the basis of quantifying physical quantities and enable us to communicate these quantities effectively.

Units can measure different types of quantities like length, area, volume, mass, and much more. Understanding the various units of measurement and how to convert them is essential for comparing and calculating real-world scenarios.

Units such as:

• Inches, feet, and miles typically measure length in the imperial system.
• Cubic centimeters and liters are used more frequently within the metric system to measure volume.
• Square meters and hectares are common for measuring land area.

Grasping this unit language fosters clear communication and accurate scientific and everyday endeavors.