To analyze the function \( r = \sqrt{\frac{V}{\pi}} \), we can deduce several properties based on the formula used to calculate the radius of a cylinder given its volume.
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Increasing Function: As the volume \( V \) increases, the radius \( r \) also increases. This is true because the square root function is an increasing function for non-negative values. Therefore, the first response is correct: "The function is an increasing function, so as the volume of the cylinder increases, the length of the radius of the cylinder increases but not at a constant rate."
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Domain Limitations: The volume \( V \) must be greater than or equal to 0 for the radius \( r \) to be defined in the context of physical cylinders. The domain of the function is \( V \geq 0 \). Thus, the statement regarding the domain being \( V \leq 0 \) is incorrect.
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Rate of Increase: The relationship between \( V \) and \( r \) is such that as \( V \) increases, \( r \) increases, but the increase in \( r \) becomes less pronounced as \( V \) gets larger. This means the function increases at a decreasing rate, which contrasts with the suggestion that it increases at an increasing rate.
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Minimum Point: When the volume \( V \) is 0, the formula gives \( r = 0 \). Therefore, the statement about having a minimum point at (0, 0) indicating that the radius is 10 inches when the volume is 0 cubic inches is also incorrect.
Based on the above analysis, the first statement is the most accurate reflection of the key features of the function, while the others include inaccuracies related to the domain and behavior of the function.
In summary:
- The correct response is: "The function is an increasing function, so as the volume of the cylinder increases, the length of the radius of the cylinder increases but not at a constant rate."