To analyze the function \( r = \sqrt[3]{\frac{3V}{\pi}} \), we can start by recognizing that it relates the radius \( r \) of a cone to its volume \( V \).
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Increasing Function: The function is a cube root function. Since cubic roots are defined for all real numbers and are increasing, this means as the volume \( V \) increases, the radius \( r \) also increases.
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Rate of Increase: The rate at which \( r \) increases relative to \( V \) is not constant. The cube root function increases at a decreasing rate because the derivative of \( r \) with respect to \( V \) shows that as \( V \) becomes larger, the increase in \( r \) gets smaller (the slope of the curve decreases).
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Minimum Point: The minimum point at \( (0, 0) \) indicates that when the volume \( V \) is 0, the radius \( r \) is also 0. However, there is no indication in the question that when the volume is 10 cubic feet, the radius is 0; rather, this statement is incorrect. It should state that when \( V \) is 0, \( r \) is 0.
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Range of r: The range of radius \( r \) is non-negative (i.e., \( r \geq 0 \)). Negative values cannot be considered for a physical radius.
Based on this analysis, the correct choice would be aligned with these features. Among the provided responses, the first option correctly reflects that the function is increasing at a decreasing rate.
Correct Response: "The function is an increasing function, so the length of the radius of the cone is increasing at a decreasing rate."