A rectangular plot of land has sides with lengths of 38 m and 52 m corrected to the nearest m. Find the range…

1. Analyze the Question

We are given the dimensions of a rectangular plot as 38 m and 52 m, both corrected to the nearest meter. We need to find the range of possible values for the area ($A$) of the rectangle.

2. Explain the Concept

When a measurement is given to the nearest whole number (1 m), the maximum error is half of that unit.

• Maximum error = $\frac{1}{2} \times 1\text{ m} = 0.5\text{ m}$.
• The lower bound of a measurement is: $\text{Value} - 0.5$.
• The upper bound of a measurement is: $\text{Value} + 0.5$.

For a rectangle with length $L$ and width $W$, the area $A = L \times W$.

• The minimum area is calculated using the lower bounds: $A_{min} = L_{min} \times W_{min}$.
• The maximum area is calculated using the upper bounds: $A_{max} = L_{max} \times W_{max}$.

3. Show the Solution

Step 1: Determine the bounds for the sides.

• For the side 38 m: • Lower bound ($L_1$) = $38 - 0.5 = 37.5\text{ m}$ • Upper bound ($U_1$) = $38 + 0.5 = 38.5\text{ m}$
• For the side 52 m: • Lower bound ($L_2$) = $52 - 0.5 = 51.5\text{ m}$ • Upper bound ($U_2$) = $52 + 0.5 = 52.5\text{ m}$

Step 2: Calculate the minimum and maximum area.

• Minimum Area ($A_{min}$) = $37.5 \times 51.5$ • $37.5 \times 51.5 = 1931.25\text{ m}^2$
• Maximum Area ($A_{max}$) = $38.5 \times 52.5$ • $38.5 \times 52.5 = 2021.25\text{ m}^2$

Step 3: Express the range. The area $A$ must be greater than or equal to the minimum area and strictly less than the maximum area (since the upper bound is the limit the value approaches). Range: $1931.25\text{ m}^2 \leq A < 2021.25\text{ m}^2$

4. Identify the Answer

Comparing our result with the given options, it matches Option A.

5. Tip

Always remember the "Half-Unit Rule": If a value is given to the nearest $n$, the range is always $\pm \frac{n}{2}$. If you are asked for the area, never just multiply the given numbers; always calculate the bounds first!