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When recording live performances, sound engineers often use a microphone with a cardioid pickup...

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When recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. Suppose the microphone is placed 4m from the front of the stage and the boundary of the optimal pickup region is given by the cardioid {eq}r = 8 + 8 \sin {/eq} 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 microscope. Answer their question and round to two decimal places. _____ {eq}(m^2) {/eq}

Area inside Cardioid in Polar Form:

{eq}\\ {/eq}

Application of Integration is used to find the area inside a cardioid.

The area inside the cardioid in polar form is given by the formula :-

{eq}\boxed{A=\displaystyle \int_{a }^{b }\dfrac{1}{2}r^2d\theta} {/eq}, where a, b are angles measured in radians and may be defined as the limits of integration.

Answer and Explanation: 1

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{eq}\\ {/eq}

Given : Equation of cardioid {eq}r= 8+8 \sin \theta {/eq}

As the sound receiving area covers the four quadrants, therefore the limits...

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Cardioid in Math: Definition, Equation & Examples

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Chapter 1 / Lesson 13
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This lesson will cover a neat shape studied in upper-level mathematics called a cardioid. We will look at the basic shape, how it is constructed, its equation in polar form, and various examples of these equations and corresponding cardioids.


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