\begin{tabular}{l|l} Sand falls from a conveyor belt at a & \( \begin{array}{l}\text { The height is changing at a } \\ \text { rate of } 11 \mathrm{~m}^{3} / \text { min onto the top of a }\end{array} \) \\ rate of \( \square \mathrm{cm} / \mathrm{min} \) when the \\ conical pile. The height of the pile is & height is 8 m . \\ always three eighths of the base & (Type an exact answer, \\ diameter. How fast are the height & using \( \pi \) as needed.) \\ and the radius changing when the & \\ pile is 8 m high? Answer in & \\ centimeters per minute. \end{tabular}

We are given that sand is being added so that the volume of a conical pile increases at   dV/dt = 11 m³/min. It is also given that the height of the pile is always three‐eighths of the base’s diameter. In other words,   h = (3/8) · (diameter). Since the radius is half the diameter, we have   diameter = 2r  →  h = (3/8)(2r) = (3/4)r. It is often easier to write the relation solved for r:   r = (4/3)h. The volume V of a right circular cone is   V = (1/3)πr²h. Substitute r = (4/3)h:   V = (1/3)π[(4/3)h]²·h = (1/3)π (16/9)h³ = (16π/27)h³. Differentiate both sides with respect to time t:   dV/dt = (16π/27) · 3h² (dh/dt) = (48π/27) h² (dh/dt) = (16π/9)h² (dh/dt). We are asked to find the rate of change of the height when the pile is 8 m high. So substitute h = 8 m and dV/dt = 11 m³/min:   11 = (16π/9)·(8)² · (dh/dt). Calculate (8)² = 64, so:   11 = (16π/9)·64 · (dh/dt). Multiplying 16 and 64 gives 1024 so we have:   11 = (1024π/9)(dh/dt). Now solve for dh/dt:   dh/dt = (11 · 9)/(1024π) = (99)/(1024π) m/min. Because the question asks for the answer in centimeters per minute, we convert:   1 m = 100 cm  ⇒  dh/dt = (99 · 100)/(1024π) = 9900/(1024π) cm/min. You can simplify this by dividing numerator and denominator by 2:   9900/1024 = 4950/512 = 2475/256. Thus,   dh/dt = (2475)/(256π) cm/min. Next, we find the rate at which the radius changes. Recall that   r = (4/3)h. Differentiate with respect to time:   dr/dt = (4/3)(dh/dt). Substitute the value we found for dh/dt:   dr/dt = (4/3)·(99/(1024π)) = (396)/(3072π) m/min. Simplify by dividing numerator and denominator by 12:   396 ÷ 12 = 33  and  3072 ÷ 12 = 256. So,   dr/dt = 33/(256π) m/min. Again, converting to centimeters per minute:   dr/dt = (33·100)/(256π) = 3300/(256π) cm/min. We can simplify further by dividing numerator and denominator by 4:   3300 ÷ 4 = 825  and  256 ÷ 4 = 64. Thus,   dr/dt = (825)/(64π) cm/min. So our final answers are: • The height is increasing at a rate of (2475)/(256π) centimeters per minute. • The radius is increasing at a rate of (825)/(64π) centimeters per minute. These are the exact answers (using π as needed).