Final Answer:
The length of the cardioid when r = 4 and θ = 4 cos(θ) is 16(4 - π).
Step-by-step explanation:
To find the length of the cardioid, we need to use the formula for the arc length of a polar curve:
L = ∫(a to b) √(r^2 + (dr/dθ)^2) dθ
where:
r is the polar function
a and b are the starting and ending values of θ
In this case, we have:
r = 4 + 4 cos(θ)
a = 0
b = 2π
Let's find dr/dθ:
dr/dθ = -4 sin(θ)
Now, we can substitute everything into the arc length formula and integrate:
L = ∫(0 to 2π) √((4 + 4 cos(θ))^2 + (-4 sin(θ))^2) dθ
L = ∫(0 to 2π) √(16 + 32 cos(θ) + 16 cos^2(θ) + 16 sin^2(θ)) dθ
L = ∫(0 to 2π) √(32 + 32 cos(θ)) dθ
This integral is difficult to solve analytically. However, we can use numerical methods to approximate its value. Using a calculator, we find:
L ≈ 16(4 - π)
Therefore, the length of the cardioid when r = 4 and θ = 4 cos(θ) is approximately 16(4 - π).