Problem
A circular portion is to be cut from the square carpet of area 4900 m². Find the radius and area of the largest circular portion possible from the carpet.
Solution
Step 1: Finding the side length of the square
Given that the area of the square carpet is 4900 m², we can find the side length by taking the square root:
$$\text{Area of square} = \text{side}^2 = 4900 \text{ m}^2$$
$$\text{Side length} = \sqrt{4900} = 70 \text{ m}$$
Step 2: Determining the radius of the largest circle
The largest circle that can be cut from a square is inscribed in the square, touching all four sides. Therefore, the diameter of the circle equals the side length of the square.
$$\text{Diameter of circle} = \text{Side of square} = 70 \text{ m}$$
$$\text{Radius} = \frac{\text{Diameter}}{2} = \frac{70}{2} = 35 \text{ m}$$
Step 3: Calculating the area of the circle
Using the formula for the area of a circle:
$$\text{Area} = \pi r^2$$
$$\text{Area} = \pi \times (35)^2 = \pi \times 1225 = 1225\pi \text{ m}^2$$
$$\text{Area} \approx 3.14159 \times 1225 \approx 3846.15 \text{ m}^2$$
Answer
- Radius of the largest circular portion: 35 m
- Area of the circular portion: \(1225\pi\) m² or approximately 3846.15 m²
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