Score \( 0 / 3 \) Penalty: none Question Find the value of \( \theta \) in the interval \( 0 \leq \theta \leq \frac{\pi}{2} \) that corresponds to the point on the curve \( r=-4-4 \sin (3 \theta)-2 \cos (-\theta) \) with the greatest distance from the pole. Round to the nearest thousandth.

It seems that I cannot directly solve the equation using the tools available. Instead, I will provide a numerical approach to find the value of \( \theta \) that maximizes \( r \). ### Step 6: Numerical Solution We can evaluate the function \( r(\theta) \) at several points in the interval \( 0 \leq \theta \leq \frac{\pi}{2} \) to find the maximum value. Let's evaluate \( r(\theta) \) at key points in the interval: 1. \( \theta = 0 \) 2. \( \theta = \frac{\pi}{6} \) 3. \( \theta = \frac{\pi}{4} \) 4. \( \theta = \frac{\pi}{3} \) 5. \( \theta = \frac{\pi}{2} \) ### Step 7: Calculate \( r(\theta) \) at these points 1. For \( \theta = 0 \): \[ r(0) = -4 - 4 \sin(0) - 2 \cos(0) = -4 - 0 - 2 = -6 \] 2. For \( \theta = \frac{\pi}{6} \): \[ r\left(\frac{\pi}{6}\right) = -4 - 4 \sin\left(\frac{\pi}{2}\right) - 2 \cos\left(\frac{\pi}{6}\right) = -4 - 4 - 2 \cdot \frac{\sqrt{3}}{2} = -8 - \sqrt{3} \approx -9.732 \] 3. For \( \theta = \frac{\pi}{4} \): \[ r\left(\frac{\pi}{4}\right) = -4 - 4 \sin\left(\frac{3\pi}{4}\right) - 2 \cos\left(\frac{\pi}{4}\right) = -4 - 4 \cdot \frac{\sqrt{2}}{2} - 2 \cdot \frac{\sqrt{2}}{2} = -4 - 2\sqrt{2} \approx -8.828 \] 4. For \( \theta = \frac{\pi}{3} \): \[ r\left(\frac{\pi}{3}\right) = -4 - 4 \sin(3 \cdot \frac{\pi}{3}) - 2 \cos\left(\frac{\pi}{3}\right) = -4 - 4 \cdot \sin(\pi) - 2 \cdot \frac{1}{2} = -4 - 0 - 1 = -5 \] 5. For \( \theta = \frac{\pi}{2} \): \[ r\left(\frac{\pi}{2}\right) = -4 - 4 \sin\left(\frac{3\pi}{2}\right) - 2 \cos\left(\frac{\pi}{2}\right) = -4 - 4 \cdot (-1) - 0 = -4 + 4 = 0 \] ### Step 8: Compare values Now we compare the values of \( r(\theta) \): - \( r(0) \approx -6 \) - \( r\left(\frac{\pi}{6}\right) \approx -9.732 \) - \( r\left(\frac{\pi}{4}\right) \approx -8.828 \) - \( r\left(\frac{\pi}{3}\right) \approx -5 \) - \( r\left(\frac{\pi}{2}\right) = 0 \) The maximum value occurs at \( \theta = \frac{\pi}{2} \). ### Final Answer Thus, the value of \( \theta \) that corresponds to the point on the curve with the greatest distance from the pole is: \[ \theta = \frac{\pi}{2} \approx 1.571 \] Rounding to the nearest thousandth, the answer is: \[ \theta \approx 1.571 \]