Problem 47 - Calculus for Biology and Medicine

Question

Experimental study plots are often squares of length \(1 \mathrm{~m}\). If \(1 \mathrm{ft}\) corresponds to \(0.305 \mathrm{~m}\), compute the area of a square plot of length \(1 \mathrm{~m}\) in \(\mathrm{ft}^{2}\).

Step-by-Step Solution

Answer

The area of the plot is approximately \(10.75 \text{ ft}^2\).

Step 1: Conversion Factor

First, identify the conversion factor between meters and feet. We know that \(1 \text{ ft} = 0.305 \text{ m}\). Our task is to convert from meters to feet, so we can write this as \(1 \text{ m} = \frac{1}{0.305} \text{ ft}\).

Step 2: Convert Length from Meters to Feet

Since the side of the square plot is 1 meter, we can convert this length to feet using the conversion factor: \(1 \text{ m} \times \frac{1}{0.305} \text{ ft/m} = \frac{1}{0.305} \text{ ft}\).

Step 3: Calculate the Area in Square Feet

To find the area of the square plot in square feet, use the formula for the area of a square: \(\text{Area} = \text{side length}^2\). Thus, the area becomes \(\left(\frac{1}{0.305}\right)^2 \text{ ft}^2\).

Step 4: Simplify the Calculation

Calculate \(\left(\frac{1}{0.305}\right)^2\) to get the area in square feet. Compute \( \frac{1}{0.305} \approx 3.27869\), and then square this number: \(3.27869^2 \approx 10.75\). Therefore, the area is approximately \(10.75 \text{ ft}^2\).

Key Concepts

Square Meters to Square Feet

Converting measurements across different units can seem daunting, but breaking it down makes it manageable. To convert square meters to square feet, we first need to understand the relationship between meters and feet.

• To convert a unit of length from meters to feet, use the conversion: \[1 \text{ m} = \frac{1}{0.305} \text{ ft} \approx 3.27869 \text{ ft}\]
• Square meters and square feet are units of area. Thus, when converting an area, we must account for the conversion in both dimensions (length and width) of the square. For a one-meter side, it becomes:\[1 \text{ m}^2 = (3.27869 \text{ ft})^2\]

Understanding this concept helps to navigate through similar conversions, ensuring accuracy in calculations.

Conversion Factor

The conversion factor is a crucial element in unit conversions. It lets us switch from one unit system to another with simple multiplication or division. Here’s how it works:

• A conversion factor expresses how many of one unit equal another. For length, if \(1 \text{ ft} = 0.305 \text{ m}\), then to convert between these, use:\[1 \text{ m} = \frac{1}{0.305} \text{ ft}\]
• When converting areas, the length conversion factor must be squared because area is a two-dimensional measurement. Thus, \[(\text{Conversion Factor for length})^2 = \text{Conversion Factor for area}\]

This principle is helpful not only for working with feet and meters but with any unit conversion task.

Area Calculation

Calculating area involves understanding the shape you are dealing with. For squares, the process is straightforward:

• First, know the formula: \[\text{Area of a square} = \text{side length}^2\]
• When given a side length in meters, and needing the area in square feet, start by converting the side length into feet.For a square of side length 1 meter:\[\text{Side length in feet} = \frac{1}{0.305} \text{ ft}\]
• Calculate the area by squaring this converted side length to find: \[(3.27869 \text{ ft})^2 = 10.75 \text{ ft}^2\]

Being meticulous with units ensures your results are consistent and precise.