In the United States, land is measured in acres and one acre is 43,560 sq ft.
a. If you buy a one-acre lot that is in the shape of a square, what would be the length of each side in feet?
b. A newspaper advertisement states that all lots in a new housing development will be a minimum of one and a half acres. Assuming the lot is rectangular and has 150 ft of frontage, how deep will the minimal-size lot be? If the new home owner wants to fence in the lot, how many yards of fencing would be needed?
c. The metric unit for measuring land is the square hectometer. (A hectometer is a length of 100 meters.) Find the size of a one-acre lot if it were measured in square hectometers.
d. A hectare is 100 acres. How many one-acre lots can fit in a square mile? How many hectares is that?
The Correct Answer and Explanation is :
a. Length of Each Side in Feet (Square Lot)
Given that 1 acre = 43,560 square feet, if the lot is square-shaped, the area of the square is 43,560 square feet. The area of a square is calculated as the side length squared, so we can use the following equation:
[
\text{Area of square} = \text{side length}^2
]
Let ( s ) be the side length of the square. We have:
[
s^2 = 43,560
]
To find ( s ), we take the square root of both sides:
[
s = \sqrt{43,560} \approx 208.71 \text{ feet}
]
So, the length of each side of the square lot would be approximately 208.71 feet.
b. Minimal Size of a Rectangular Lot and Fencing
Minimal Size of Lot:
The advertisement states that the lot will be at least 1.5 acres. The area of 1.5 acres is:
[
1.5 \text{ acres} \times 43,560 \text{ square feet/acre} = 65,340 \text{ square feet}
]
The lot has 150 feet of frontage, so to find the depth, we use the formula for the area of a rectangle:
[
\text{Area of rectangle} = \text{frontage} \times \text{depth}
]
Let ( d ) be the depth of the lot. Then:
[
150 \times d = 65,340
]
Solving for ( d ):
[
d = \frac{65,340}{150} = 435.6 \text{ feet}
]
So, the minimal depth of the lot is 435.6 feet.
Fencing Required:
The fencing will be needed for the perimeter of the rectangular lot. The perimeter ( P ) of a rectangle is given by:
[
P = 2 \times (\text{frontage} + \text{depth})
]
Substituting the values:
[
P = 2 \times (150 + 435.6) = 2 \times 585.6 = 1,171.2 \text{ feet}
]
Since fencing is usually measured in yards (3 feet per yard), we convert the length in feet to yards:
[
\text{Fencing in yards} = \frac{1,171.2}{3} \approx 390.4 \text{ yards}
]
So, the new homeowner would need approximately 390.4 yards of fencing.
c. Size of One-Acre Lot in Square Hectometers
1 acre = 43,560 square feet. To convert square feet to square meters, we use the conversion factor:
[
1 \text{ meter} = 3.28084 \text{ feet}
]
So, 1 square meter = ( (3.28084)^2 = 10.7639 \text{ square feet} ).
Now, convert 43,560 square feet to square meters:
[
\text{Area in square meters} = \frac{43,560}{10.7639} \approx 4046.86 \text{ square meters}
]
Next, to convert square meters to square hectometers:
[
1 \text{ hectometer} = 100 \text{ meters}, \quad 1 \text{ square hectometer} = 100^2 = 10,000 \text{ square meters}
]
Thus, the area of the one-acre lot in square hectometers is:
[
\text{Area in square hectometers} = \frac{4046.86}{10,000} \approx 0.4047 \text{ square hectometers}
]
So, the size of the one-acre lot in square hectometers is approximately 0.4047 square hectometers.
d. One-Acre Lots in a Square Mile and Hectares
1 square mile = 640 acres. Since 1 hectare = 100 acres, we can calculate how many hectares fit in a square mile:
[
\text{Hectares in a square mile} = \frac{640}{100} = 6.4 \text{ hectares}
]
To determine how many one-acre lots fit in a square mile, we simply divide the number of acres in a square mile by the number of acres in each lot:
[
\text{One-acre lots in a square mile} = 640 \text{ acres} \div 1 \text{ acre/lot} = 640 \text{ lots}
]
Therefore, there are 640 one-acre lots in a square mile.
To summarize:
- A square mile contains 640 one-acre lots.
- A square mile is equivalent to 6.4 hectares.
This breakdown shows the importance of understanding how units of land measurement convert into one another, whether in imperial or metric systems. By converting between different units, you can calculate dimensions, areas, and other necessary details for land development or property management.