Verified Answer :
- Calculate the total area in hectares: \(\frac{30}{2.5} = 12 \text{ hectares}\).
- Convert the area to square meters: \(12 \times 10,000 = 120,000 \text{ m}^2\).
- Define the relationship between length (\(l\)) and width (\(w\)): \(l \times w = 120,000\).
- Conclude that there are infinitely many dimensions satisfying \(\boxed{l = \frac{120,000}{w}}\).
Explanation
- Calculating the Total Area in Hectares
To determine the dimensions of the land, we first calculate the total area the given amount of grass seed can cover. We are given:
- The seeding rate: \(2.5\) kg/hectare.
- The total amount of seed: \(30\) kg.
Using the relationship \(\text{Area (hectares)} = \frac{\text{Total seed}}{\text{Seeding rate}}\), we find the area in hectares:
\[h = \frac{30}{2.5} = 12 \text{ hectares}.\]
- Converting Area to Square Meters
Since the dimensions of the land are requested in meters, we must convert the area from hectares to square meters. We know that \(1\) hectare is equivalent to \(10,000\) square meters. Thus, the total area \(A\) in square meters is:
\[A = 12 \times 10,000 = 120,000 \text{ m}^2.\]
- Forming the Equation and Determining Dimensions
Let \(l\) represent the length of the land in meters and \(w\) represent the width in meters. The area of a rectangular plot is given by the product of its length and width:
\[l \times w = 120,000.\]
This is the required equation. Because there are no further constraints given (such as a specific ratio between the length and width or a fixed value for one of the dimensions), there are infinitely many possible pairs of \((l, w)\) that satisfy this relationship. For example, if the width \(w\) were \(200\) meters, the length would be:
\[l = \frac{120,000}{200} = 600 \text{ meters}.\]
Thus, the dimensions can be expressed by any pair \((l, w)\) such that \(l = \frac{120,000}{w}\).
Examples
In real-life land management, such as setting up agricultural plots or landscaping projects, determining dimensions based on resource availability is a common task. For instance, if you have a limited amount of fertilizer, you must calculate the total area you can treat, and then design your plot layout accordingly to ensure optimal resource coverage. This helps in efficient land planning and prevents wastage of materials.