Step 1: Identify the given information.
The cardioid is given by:
r = 8 + 8sin(θ)
The microphone is located 4 meters from the front of the stage.
We want to find the area inside the cardioid that is also on the stage.
Step 2: Determine whether the cardioid extends beyond the stage.
Since y = r sin(θ), substitute the cardioid equation:
y = (8 + 8sin(θ))sin(θ)
y = 8sin(θ) + 8sin²(θ)
Complete the square:
y = 8(sin(θ) + 0.5)² - 2
The minimum value of y is -2 meters.
Since the microphone is 4 meters from the front of the stage, the cardioid extends only 2 meters toward the front. Therefore, the entire cardioid lies on the stage.
Step 3: Calculate the area of the cardioid.
The polar area formula is:
A = (1/2) ∫ r² dθ
Substitute r = 8 + 8sin(θ):
A = (1/2) ∫₀²π (8 + 8sin(θ))² dθ
Expand:
A = 32 ∫₀²π (1 + 2sin(θ) + sin²(θ)) dθ
Using the standard trigonometric integrals:
∫₀²π 1 dθ = 2π
∫₀²π 2sin(θ) dθ = 0
∫₀²π sin²(θ) dθ = π
Therefore:
A = 32(2π + 0 + π)
A = 96π
A ≈ 301.5929
Final answer: 301.59 square meters.
