- Determine the total area: \(30 \text{ kg} / 2.5 \text{ kg/ha} = 12 \text{ ha} = 120,000 \text{ m}^2\).
- Define the area equation: \(l \cdot w = 120,000\).
- Recognize that \(l\) and \(w\) are not uniquely determined without further constraints.
- Express the length as a function of width: \(l = \boxed{\frac{120,000}{w}}\).
Explanation
- Determining the Total Area
The project uses 2.5 kg of grass seed per hectare and has a total of 30 kg of seed available. To find the total area that can be seeded, we divide the total available seed by the seed consumption rate per hectare:
\[ \text{Area (in hectares)} = \frac{30 \text{ kg}}{2.5 \text{ kg/ha}} = 12 \text{ hectares}. \]
Since one hectare is equivalent to 10,000 square meters, we convert this area into square meters to maintain consistent units for the length (\(l\)) and width (\(w\)) measured in meters:
\[ \text{Area (in square meters)} = 12 \text{ hectares} \times 10,000 \frac{\text{m}^2}{\text{hectare}} = 120,000 \text{ m}^2. \]
- Formulating the Equation for Area
The area \(A\) of a rectangular piece of land is defined by the product of its length \(l\) and its width \(w\). Given that the area is 120,000 \(\text{m}^2\), we can formulate the following equation:
\[ l \cdot w = 120,000. \]
This equation represents the relationship between the length and width of the land based on the available grass seed.
- Solving for Length and Width
We have one equation with two unknown variables, \(l\) and \(w\). A system with more variables than independent equations generally does not have a unique solution unless additional information (such as the ratio of length to width or the perimeter of the land) is provided. Without such constraints, we can express the length in terms of the width as:
\[ l = \frac{120,000}{w}. \]
For example, if the width \(w\) were 300 meters, the length \(l\) would be \(\frac{120,000}{300} = 400\) meters. Conversely, if the width were 200 meters, the length would be \(\frac{120,000}{200} = 600\) meters. Because no further specific information is given, we cannot determine a single pair of values for \(l\) and \(w\).
Examples
Imagine a landscape architect planning to plant wildflowers in a community park. If they are allocated a specific budget that limits them to purchasing enough seed for exactly 120,000 square meters of ground, they must decide on the dimensions of the rectangular plot. Knowing that the area is fixed, they can adjust the width based on the terrain or accessibility and then calculate the necessary length to ensure they stay within the budget constraints. This ensures efficient resource management and proper coverage of the designated area.