To find the amount of liner needed to cover the bottom and all four sides of the rectangular swimming pool, we need to calculate the total surface area of the bottom and the sides of the pool.
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Area of the bottom of the pool: The bottom area of the pool can be calculated as follows: \[ \text{Area of the bottom} = \text{length} \times \text{width} = 10 , \text{m} \times 5 , \text{m} = 50 , \text{m}^2 \]
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Area of the sides of the pool: The pool has four sides, and we will calculate the area of each side.
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Two longer sides (length = 10 m, height = 2 m): \[ \text{Area of one longer side} = \text{length} \times \text{height} = 10 , \text{m} \times 2 , \text{m} = 20 , \text{m}^2 \] Total area of two longer sides: \[ 2 \times 20 , \text{m}^2 = 40 , \text{m}^2 \]
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Two shorter sides (width = 5 m, height = 2 m): \[ \text{Area of one shorter side} = \text{width} \times \text{height} = 5 , \text{m} \times 2 , \text{m} = 10 , \text{m}^2 \] Total area of two shorter sides: \[ 2 \times 10 , \text{m}^2 = 20 , \text{m}^2 \]
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Total area of the sides: Adding the areas of the longer sides and shorter sides: \[ \text{Total area of sides} = 40 , \text{m}^2 + 20 , \text{m}^2 = 60 , \text{m}^2 \]
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Total area to be covered by the liner: Finally, we add the area of the bottom and the total area of the sides: \[ \text{Total area} = \text{Area of bottom} + \text{Total area of sides} = 50 , \text{m}^2 + 60 , \text{m}^2 = 110 , \text{m}^2 \]
Therefore, the total amount of liner needed is \(\boxed{110 , \text{m}^2}\).