Question

identify and graph the polar equation.
r = 7 + 14 sinθ
what shape does the polar equation form when graphed?
a. limacon without an inner loop.
b. cardioid that is symmetric to the polar axis.
c. limacon with an inner loop.
d. cardioid that is symmetric to line θ = π/2
which graph represents the equation?

Explanation:

Step1: Recall the general form of polar equations

The general form of a limacon is \(r = a + b\sin\theta\) or \(r=a + b\cos\theta\). For a cardioid, \(|a|=|b|\). Here, \(a = 7\) and \(b = 14\), so it's a limacon.

Step2: Determine the type of limacon

We use the ratio \(\frac{a}{b}\). Given \(a = 7\) and \(b = 14\), \(\frac{a}{b}=\frac{7}{14}=\frac{1}{2}<1\). When \(\frac{a}{b}<1\) for \(r=a + b\sin\theta\), the limacon has an inner loop.

Step3: Check symmetry

For \(r = 7+14\sin\theta\), we use the symmetry test about the line \(\theta=\frac{\pi}{2}\). Replace \((r,\theta)\) with \((r,\pi - \theta)\): \(r=7 + 14\sin(\pi-\theta)=7 + 14\sin\theta\) (since \(\sin(\pi-\theta)=\sin\theta\)). So it's symmetric about the line \(\theta=\frac{\pi}{2}\)

Answer:

C. Limacon with an inner loop.