Find the length of the curve.
The cardioid r equals 3 left parenthesis 1 minus sine theta right parenthesis
Question content area bottom
Part 1
A.StartFraction 27 Over 2 EndFraction
StartFraction 27 Over 2 EndFraction
B.StartFraction 27 Over 2 EndFraction pi
StartFraction 27 Over 2 EndFraction
pi
C.24
24
D.24 pi
24pi

To find the length of the curve for the cardioid \( r = 3(1 - \sin \theta) \), we use the formula for the length of a curve in polar coordinates:

\[
L = \int_{\alpha}^{\beta} \sqrt{ \left( \frac{dr}{d\theta} \right)^2 + r^2 } \, d\theta
\]

### Step 1: Find \(\frac{dr}{d\theta}\)
Given \( r = 3(1 - \sin \theta) \):

\[
\frac{dr}{d\theta} = 3(-\cos \theta)
\]

### Step 2: Substitute into the Length Formula
Now, we substitute \( r \) and \( \frac{dr}{d\theta} \) into the formula:

\[
L = \int_{0}^{2\pi} \sqrt{ (3(-\cos \theta))^2 + (3(1 - \sin \theta))^2 } \, d\theta
\]

Simplifying inside the square root:

\[
L = \int_{0}^{2\pi} \sqrt{ 9\cos^2 \theta + 9(1 - \sin \theta)^2 } \, d\theta
\]

\[
L = \int_{0}^{2\pi} \sqrt{ 9\cos^2 \theta + 9(1 - 2\sin \theta + \sin^2 \theta) } \, d\theta
\]

Factor out the 9:

\[
L = \int_{0}^{2\pi} \sqrt{ 9(\cos^2 \theta + 1 - 2\sin \theta + \sin^2 \theta) } \, d\theta
\]

\[
L = 3\int_{0}^{2\pi} \sqrt{ \cos^2 \theta + \sin^2 \theta - 2\sin \theta + 1 } \, d\theta
\]

\[
L = 3\int_{0}^{2\pi} \sqrt{ 1 - 2\sin \theta + 1 } \, d\theta
\]

\[
L = 3\int_{0}^{2\pi} \sqrt{ 2(1 - \sin \theta) } \, d\theta
\]

\[
L = 3\sqrt{2} \int_{0}^{2\pi} \sqrt{ 1 - \sin \theta } \, d\theta
\]

This integral can be computed using standard techniques, which leads to:

\[
L = 24
\]

So, the correct answer is:

\[
\boxed{C. \, 24}
\]

Would you like more details or have any questions? Here are some follow-up questions to consider:

1. What are the standard integrals in polar coordinates for curve length?
2. How does the symmetry of a curve affect the calculation of its length?
3. How would the length calculation change if \( r \) was given as a function of cosine?
4. What is the significance of the cardioid shape in polar coordinates?
5. How do you calculate the area enclosed by a cardioid?

**Tip:** Always check for symmetries in polar curves to simplify calculations.

Learn how to find the length of the curve for the cardioid r = 3(1 - sin θ) using polar coordinates. Detailed step-by-step solution provided, including integration techniques and simplification. Suitable for advanced high school students or anyone studying calculus.

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