Solving Rectangle Perimeter and Area Problem
This problem asks us to find the side length of a square whose area is the same as a given rectangle. We are provided with the rectangle's perimeter and the difference between its length and width.
Rectangle Dimensions Calculation
Let's denote the length of the rectangle as '$l$' and the width as '$w$'.
- We are given the perimeter of the rectangle is 280 m. The formula for the perimeter of a rectangle is $P = 2(l + w)$. So, we have the equation: $$2(l + w) = 280 \text{ m}$$ Dividing both sides by 2, we get: $$l + w = 140 \text{ m} \quad (1)$$
- We are also given that the difference between its two sides is 40 m. Assuming the length is greater than the width, we can write: $$l - w = 40 \text{ m} \quad (2)$$
- Now we have a system of two linear equations with two variables. We can solve this system to find the values of '$l$' and '$w$'. Add equation (1) and equation (2): $$(l + w) + (l - w) = 140 + 40$$ $$2l = 180$$ $$l = \frac{180}{2}$$ $$l = 90 \text{ m}$$
- Substitute the value of '$l$' back into equation (1): $$90 + w = 140$$ $$w = 140 - 90$$ $$w = 50 \text{ m}$$
So, the dimensions of the rectangle are 90 m and 50 m.
Rectangle Area Calculation
The area of a rectangle is calculated by multiplying its length and width ($A = l \times w$). Using the dimensions we found:
$$A_{\text{rectangle}} = 90 \text{ m} \times 50 \text{ m}$$ $$A_{\text{rectangle}} = 4500 \text{ m}^2$$Square Side Calculation
The problem states that the area of a square is equal to the area of this rectangle. Let the side length of the square be '$s$'. The area of a square is given by $A_{\text{square}} = s^2$. Therefore:
$$s^2 = A_{\text{rectangle}}$$ $$s^2 = 4500 \text{ m}^2$$To find the side length '$s$', we need to take the square root of the area:
$$s = \sqrt{4500 \text{ m}^2}$$Now, let's simplify the square root:
$$s = \sqrt{45 \times 100} \text{ m}$$ $$s = \sqrt{9 \times 5 \times 100} \text{ m}$$We can take the square root of the perfect squares (9 and 100):
$$s = \sqrt{9} \times \sqrt{5} \times \sqrt{100} \text{ m}$$ $$s = 3 \times \sqrt{5} \times 10 \text{ m}$$ $$s = 30\sqrt{5} \text{ m}$$Final Answer
The side of the square whose area is equal to the area of the rectangle is $30\sqrt{5}$ m.
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