Problem 81 - Elementary Algebra Within Reach

Question

Geometry A rectangular plot of land measures 650 feet by 825 feet. A square garage with side lengths of 24 feet is built on the plot of land. What percent of the plot of land is occupied by the garage?

Step-by-Step Solution

Answer

The garage occupies approximately 0.071% of the total land.

Step 1: Calculate Total Land Area

We calculate the total area of the plot of land by multiplying the length (825 feet) by the width (650 feet). In mathematical terms, that is \(Area_{total} = 825ft \times 650ft\).

Step 2: Calculate Garage Area

We calculate the total area of the square garage by squaring the length of its sides (24 feet). Mathematically, this is represented as \(Area_{garage} = 24ft \times 24ft\).

Step 3: Calculate Percentage

We then calculate how much of the total land area the garage uses. This is done by dividing the area of the garage by the total area of the land and then multiplying by 100 to get the value in percent. Mathematically, this is represented as \(\% Area_{used} = \left( \frac{Area_{garage}}{Area_{total}} \right) \times 100\% \).

Key Concepts

Rectangular Plot Area Calculation

Understanding the geometry of a rectangular plot is crucial when dealing with land area calculations. To find the area of a rectangle, we simply multiply its length by its width. The formula is represented as:

\(Area_{rectangle} = length \times width\).

In the context of our exercise, the length and width are given as 825 feet and 650 feet respectively. Therefore, applying the formula:
\(Area_{total} = 825 \text{ ft} \times 650 \text{ ft}\), we obtain the total area of the plot. It's important here to ensure that both measurements are in the same units, feet in this case, to obtain a consistent result.

The formula is straightforward, yet precise calculations are essential as any errors can significantly impact the overall area assessment, especially for large plots.

Square Area Calculation

Squares have the interesting property that all sides are equal in length. To compute the area of a square, we need to use the formula:

\(Area_{square} = side \times side\) or simply \(Area_{square} = side^2\).

When looking at our example of a garage built on the plot of land, where each side of the garage is 24 feet, we apply the second form of the formula:
\(Area_{garage} = 24 \text{ ft} \times 24 \text{ ft}\) or \(Area_{garage} = 576 \text{ ft}^2\). It's the simplicity of this formula that makes calculations involving squares quick and unambiguous. One should note that because of the geometry of the square, all areas calculated will be perfect squares numerically.

Percentage Calculation

Percentage calculations are an integral part of comparing quantities. To find out what percentage one value is of another, we use the following formula:

\(\text{Percentage} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100\%\).

Here, the 'Part' is the area of the garage, and the 'Whole' is the total plot area. After calculating both areas as outlined in previous sections, we divide the area of the garage by the total area of the land and multiply by 100 to convert it into a percentage.

This calculation helps us understand the extent to which the garage occupies the land. In the given exercise, this final step puts the size of the garage in perspective relative to the entire plot, which is an effective way to comprehend spatial relationships and the efficient utilization of space.