When granular materials fall, they form conical piles. The angle of slope, measured from the horizontal, at which the material comes to rest is called the angle of repose. The angle of repose is related to the height h and base radius r of the conical pile by the equation \( \theta=\cot ^{-1} \frac{\mathrm{r}}{\mathrm{h}} \). When certain granular materials are stored in a pile 12 feet high, the diameter of the base of the pile is 65 feet. (a) Find the angle of repose for these granular materials. \( \theta=20.27^{\circ} \) (Round to two decimal places as needed.) (b) What is the base diameter of a pile that is 17 feet high? diameter \( =\square \) feet (Round to two decimal places as needed.)

To find the angle of repose for the granular materials in a conical pile, we first need to calculate the radius \( r \) (half of the diameter) and the height \( h \) of the pile. Given the height \( h = 12 \) feet and the diameter \( d = 65 \) feet, we have: \( r = \frac{d}{2} = \frac{65}{2} = 32.5 \) feet. Using the formula \( \theta = \cot^{-1} \left( \frac{r}{h} \right) \): \[ \theta = \cot^{-1} \left( \frac{32.5}{12} \right) \approx 20.27^\circ. \] Now, for the second part, to find the base diameter of a pile that is 17 feet high, we can use the same angle of repose. Since we already know \( \theta \approx 20.27^\circ \), we need to find the radius based on the height \( h = 17 \) feet. From the cotangent ratio, we have: \[ \tan(\theta) = \frac{h}{r} \Rightarrow r = \frac{h}{\tan(\theta)}. \] Calculating \( r \): \[ r = \frac{17}{\tan(20.27^\circ)} \approx 17 / 0.372 \approx 45.7 \text{ feet}. \] Thus, the diameter is: \[ d = 2r \approx 2 \times 45.7 = 91.4 \text{ feet}. \] So, the base diameter of a pile that is 17 feet high is approximately \( 91.4 \) feet.