The sides of a triangular field are 41 m, 40 m and 9 m. The number of rose beds that can be prepared in the field, if each rose bed, on an average, needs 900 cm^2 space, is:

Question

Accepted Answer

To determine the number of rose beds that can be prepared in the triangular field, we first need to calculate the area of the triangle. The sides of the triangle are given as 41 m, 40 m, and 9 m. We can use Heron's formula to find the area of the triangle.

**Step 1: Calculate the semi-perimeter (s) of the triangle.**

The semi-perimeter $ s $ is calculated as follows:

$$ s = \frac{a + b + c}{2} $$

where $ a = 41 \, \text{m} $, $ b = 40 \, \text{m} $, and $ c = 9 \, \text{m} $.

$$ s = \frac{41 + 40 + 9}{2} = \frac{90}{2} = 45 \, \text{m} $$

**Step 2: Calculate the area (A) using Heron's formula.**

Heron's formula for the area $ A $ of a triangle is:

$$ A = \sqrt{s(s-a)(s-b)(s-c)} $$

Substituting the values:

$$ A = \sqrt{45(45-41)(45-40)(45-9)} $$
$$ A = \sqrt{45 \times 4 \times 5 \times 36} $$
$$ A = \sqrt{32400} $$
$$ A = 180 \, \text{m}^2 $$

**Step 3: Convert the area from square meters to square centimeters.**

Since 1 m² = 10,000 cm², the area in square centimeters is:

$$ 180 \, \text{m}^2 = 180 \times 10,000 \, \text{cm}^2 = 1,800,000 \, \text{cm}^2 $$

**Step 4: Calculate the number of rose beds.**

Each rose bed requires 900 cm². Therefore, the number of rose beds that can be prepared is:

$$ \text{Number of rose beds} = \frac{1,800,000 \, \text{cm}^2}{900 \, \text{cm}^2} = 2000 $$

Thus, the correct answer is Option 1: 2000.

**Explanation for Incorrect Options:**

- **Option 2 (1800):** This option is incorrect because the calculation shows that the area of the field can accommodate 2000 rose beds, not 1800.
  
- **Option 3 (900):** This option is incorrect because it significantly underestimates the number of rose beds that can fit in the given area.

- **Option 4 (800):** This option is incorrect as it also underestimates the capacity of the field for rose beds.

In conclusion, the calculations clearly support Option 1 as the correct answer, with the field accommodating 2000 rose beds.