Written Homework 9 Due at the beginning of class on Friday, March 27. Calculus II - Miles Note: All work should be done by hand. Of course, you may use Mathematica to check your answers. (1) (relates to 10.3, 10 points) Draw a nicely labelled sketch of the curve with polar equation r = 1 + 2 cos(θ/2). Your work and exposition should make it clear how you arrived at your graph. (2) (relates to 10.4, 20 points) When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it surpresses noise from the audience. See for example, http://www.sweetwater.com/insync/studio-microphone-buying-guide/ (about half-way down). Suppose the microphone is placed 4 m from the front of the stage (as in the figure below) and the boundary of the optimal pickup region is given by the cardioid r = 8 + 8 sin(θ), where r is measured in meters and the microphone is at the pole. The musicians want to know the area they will have on stage within the optimal pickup range of the microphone. Answer their question. Notes for problem (2): • Let α be the angle between the axis determined by the microphone and the corner of the microphone’s optimal region on the stage (as shown in the picture). You should find α exactly (it will involve an “arc-trig” function). – As a reference to check your answer, α is approximately 0.37 radians. • Computing the definite integral that gives the area of the region yields an expression involving trig functions and α. Use a triangle to eliminate the trig functions (think back to trig-sub problems) so that your final answer is a polynomial in α. – As a reference to check your answer, the area is approximately 204.16 m2 . • The following formulas may be useful: · sin2 (θ) = (1/2)(1 − cos(2θ)) · cos2 (θ) = (1/2)(1 + cos(2θ)) · 2 cos(θ) sin(θ) = sin(2θ) 1