Problem 74 - Prealgebra

Question

Marianne needs to create a rectangular garden plot covering 223 square meters \(\left(223 \mathrm{~m}^{2}\right)\). If the width of the plot is \(8.3\) meters, find the length of the plot correct to the nearest tenth of a meter.

Step-by-Step Solution

Answer

The length of the rectangular garden is 26.9 meters.

Step 1: Understand the Area Formula for a Rectangle

The area of a rectangle is calculated using the formula \( A = l \times w \), where \( A \) is the area, \( l \) is the length, and \( w \) is the width. Here, \( A = 223 \, \text{m}^2 \) and \( w = 8.3 \, \text{m} \). We need to find \( l \).

Step 2: Rearrange the Formula to Solve for Length

Rearrange the area formula to solve for \( l \): \( l = \frac{A}{w} \). Substitute the known values to find \( l \).

Step 3: Substitute the Known Values into the Formula

Substitute \( A = 223 \) and \( w = 8.3 \) into the formula: \( l = \frac{223}{8.3} \).

Step 4: Calculate the Length

Perform the division \( l = \frac{223}{8.3} \) to find the length of the garden. This gives \( l \approx 26.8674698795 \).

Step 5: Round the Length to the Nearest Tenth

Round the calculated length \( l = 26.8674698795 \) to the nearest tenth. This results in \( l = 26.9 \).

Key Concepts

Rectangle Area

In prealgebra, understanding how to calculate the area of a rectangle is a foundational skill. The area represents the amount of space inside the rectangle and is expressed in square units. To find the area, you multiply the rectangle's length by its width, using the formula:\[A = l \times w\]where:

• \( A \) is the area
• \( l \) is the length
• \( w \) is the width

Suppose you are given the area and one side length, as in the garden problem. In that case, you can rearrange this formula to find the missing dimension. This skill is crucial for geometry problems involving rectangles. Remember to ensure that both length and width are in the same units before performing multiplication.

Unit Conversion

Unit conversion is the process of converting a measurement from one unit to another. It is vital in mathematics and everyday life, especially when dealing with various measurement systems. In the context of our exercise, the units are straightforward, since both input and needed values are in meters and square meters, avoiding conversion issues. However, if you encountered a problem involving different units—say, inches for width and square feet for area—you would need to convert one set of units to match the others. This often uses conversion factors, like 1 meter equals 100 centimeters. Always check your units before solving any problem to avoid inaccuracies.

Rounding Numbers

Rounding numbers aims to make them simpler and easier to work with, especially in cases where exact value is unnecessary. Here, the garden's length is calculated as approximately \( 26.8674698795 \) meters, but for practical purposes, rounding to the nearest tenth provides a more manageable and meaningful figure.When rounding to the nearest tenth, look at the number in the hundredths place:

• If it's 5 or greater, round up.
• If it's less than 5, round down.

Thus, \( 26.8674698795 \) rounds to \( 26.9 \). When rounding, you reduce the number of digits, which can help simplify calculations and make the numbers easier to interpret or communicate.