a microphone detects sound based on the distance and direction of the sound from the microphone. the…

a microphone detects sound based on the distance and direction of the sound from the microphone. the location of the microphone is considered the origin of the polar coordinate system. the microphone detects sound horizontally, measured in feet (ft), within the graph of the polar function ( r = f(\theta) ), where ( f(\theta)=1+sin\theta ). for ( \frac{pi}{6}leq\thetaleq\frac{pi}{3} ), what is the maximum distance from the microphone that sound will be detected?\na 1.500 ft\nb 1.707 ft\nc 1.866 ft\nd 2.000 ft

a microphone detects sound based on the distance and direction of the sound from the microphone. the location of the microphone is considered the origin of the polar coordinate system. the microphone detects sound horizontally, measured in feet (ft), within the graph of the polar function ( r = f(\theta) ), where ( f(\theta)=1+sin\theta ). for ( \frac{pi}{6}leq\thetaleq\frac{pi}{3} ), what is the maximum distance from the microphone that sound will be detected?\na 1.500 ft\nb 1.707 ft\nc 1.866 ft\nd 2.000 ft

Answer

Explanation:

Step1: Analyze the function (r = 1+\sin\theta)

The function (r = 1+\sin\theta) is a polar function. The maximum value of (y = \sin\theta) in a general sense is (1), but we are restricted to the interval (\frac{\pi}{6}\leq\theta\leq\frac{\pi}{3}).

Step2: Find the maximum of (\sin\theta) in the given interval

We know that the function (y=\sin\theta) is increasing on the interval (\left[-\frac{\pi}{2},\frac{\pi}{2}\right]). Since (\frac{\pi}{6}\leq\theta\leq\frac{\pi}{3}) and (\sin\theta) is increasing in this sub - interval of (\left[-\frac{\pi}{2},\frac{\pi}{2}\right]), we evaluate (\sin\theta) at the endpoints. When (\theta=\frac{\pi}{6}), (\sin\frac{\pi}{6}=\frac{1}{2}). When (\theta = \frac{\pi}{3}), (\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\approx0.866).

Step3: Calculate (r) for (\theta=\frac{\pi}{3})

Substitute (\theta=\frac{\pi}{3}) into (r = 1+\sin\theta). Then (r=1+\sin\frac{\pi}{3}). Since (\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}), we have (r = 1+\frac{\sqrt{3}}{2}). [r=\frac{2 + \sqrt{3}}{2}\approx\frac{2+1.732}{2}=\frac{3.732}{2}=1.866]

Answer:

C. (1.866) ft