Problem 174 - Introductory Chemistry Atoms First

Question

A company wants 800 square feet of carpet, but the carpet store sells only by the square meter. How many square meters does the company need to buy? \((1 \mathrm{~m}=39.37\) in. \()\)

Step-by-Step Solution

Answer

The company needs to buy approximately 74.32 square meters of carpet.

Step 1: Convert 1 foot to meters

We know that 1 ft = 12 in. Let's convert it to meters using the given conversion factor 1 m = 39.37 in. \[1\ \text{foot} = \frac{12 \text{ inches}}{1 \text{ foot}} \times \frac{1 \text{ meter}}{39.37 \text{ inches}}=\frac{12}{39.37}\ \text{meters}\]

Step 2: Calculate the conversion factor for square feet to square meters

We have found the conversion factor for feet to meters. Now, let's find the conversion factor for square feet to square meters. Make sure to square the conversion factor. \[\left(\frac{12}{39.37} \ \text{meters}\right)^2 = \frac{144}{(39.37)^2} \ \text{square meters}\]

Step 3: Convert 800 square feet to square meters

Now, we can use the conversion factor to convert 800 square feet to square meters. \[800 \ \text{square feet} \times \frac{144}{(39.37)^2} \ \text{square meters} \approx 74.32 \ \text{square meters}\] So, the company needs to buy approximately 74.32 square meters of carpet.

Key Concepts

Understanding Square Meters

Square meters are a common unit for measuring area, particularly in countries using the metric system. This unit is ideal for measuring spaces like rooms or yards. To picture a square meter, imagine a square with each side measuring one meter. That's the space it covers.

Why use square meters? They make calculations simple when working with the International System of Units (SI). Additionally, though the metric system isn't used everywhere, converting other units into square meters can lead to a clear understanding of size and space.

Exploring Square Feet

Square feet are part of the imperial system, often used in the United States and other countries not adopting the metric system fully. One square foot is the area of a square with each side measuring one foot.

They are widely used in construction and real estate, especially in the US. Knowing how to convert square feet into other units is essential when dealing with global transactions or specifications.

The Role of Conversion Factors

Conversion factors are crucial for switching between different units of measurement. They allow us to translate measurements from one system to another accurately.

For example, converting feet to meters involves a conversion factor derived from the relationship between these units. Given that 1 meter equals 39.37 inches, this allows us to find how many meters one foot equals by dividing 12 inches by 39.37 inches.

This factor becomes \( \frac{12}{39.37} \) when converting lengths. When dealing with area, like square feet to square meters, we square this factor to get the correct conversion rate.

Navigating Measurement Conversion

Measurement conversion is an essential skill when working across different measurement systems. It involves applying the right conversion factor to switch from one unit to another effectively.

In the exercise provided, converting 800 square feet to square meters required understanding both the initial conversion of feet to meters and then squaring that to handle the area conversion. For example:

• Convert feet to meters: Use the factor \( \frac{12}{39.37} \).
• Square this factor for area: \( \left(\frac{12}{39.37}\right)^2 \).
• Apply this to 800 square feet to estimate the equivalent in square meters, yielding approximately 74.32 square meters.

This process is vital in various fields, ensuring that measurements are consistent and accurate across different systems.