To analyze the function \( w + 60x \) in the context of filling a community swimming pool, we need to clarify the variables and their units.
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Variable x: Since the pool is being filled at a rate of 60 gallons per minute, a reasonable unit for the variable \( x \) is minutes. This variable represents the time in minutes that has passed since the pool started filling.
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Variable w: This variable \( w \) represents the initial amount of water in the pool, which is starting from an empty pool. A reasonable unit for \( w \) is gallons. Since they are beginning with an empty pool, \( w \) would typically be 0 gallons at the start (when \( x = 0 \)).
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Domain: The domain of the function represents the possible values for \( x \), which is the time in minutes. Since the pool is being filled starting from an empty state and can continue to be filled until it reaches capacity, the domain can be expressed as [0, T], where \( T \) is the total time it takes to fill the pool to its maximum capacity.
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Range: The range of the function represents the possible values for \( w + 60x \), which correlates to the total volume of water in the pool as time progresses. The minimum value of the pool's water is 0 gallons (at \( x = 0 \)), and the maximum value corresponds to the total capacity of the pool when it's full, which can be expressed as [0, C], where \( C \) is the total capacity of the pool in gallons.
In summary:
- A reasonable unit for the variable \( x \) is minutes.
- A reasonable unit for the variable \( w \) is gallons.
- The domain represents the time in minutes since filling started.
- The range represents the amount of water in gallons in the pool.
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