https://file.aiquickdraw.com/mathbot/file/1726764267107tdxvep6u.jpg

Let's analyze the three mathematical problems from the image:

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### Problem 8:
**Show that the arc of the upper half of the curve \( r = a(1 - \cos \theta) \) for \( 0 \leq \theta \leq \pi \) is bisected at \( \theta = \frac{2\pi}{3} \).**

This problem involves finding the arc length of the upper half of a cardioid (a specific polar curve). The goal is to show that the arc from \( \theta = 0 \) to \( \theta = \pi \) is divided into two equal parts at \( \theta = \frac{2\pi}{3} \).

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### Problem 9:
**Determine the perimeter of the curve \( r = a \sin^3 \left( \frac{\theta}{3} \right) \), where \( 0 \leq \theta \leq 3\pi \).**

In this problem, you need to compute the perimeter (or length) of a polar curve. The perimeter of a polar curve can be determined using the formula for arc length in polar coordinates:
\[
L = \int_{\theta_1}^{\theta_2} \sqrt{r^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta
\]
We would apply this to the given function \( r = a \sin^3 \left( \frac{\theta}{3} \right) \).

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### Problem 10:
**Find the length of the spiral \( r = e^{\alpha \theta} \) from the pole to the point \( (r, \theta) \).**

This problem involves finding the length of an Archimedean spiral, where \( r \) increases exponentially with \( \theta \). The length of a spiral in polar coordinates can be determined using the arc length formula in polar coordinates.

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Would you like to proceed with solving any specific one of these problems? I can provide detailed solutions. Let me know if you'd like further clarification.

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### 5 Related Questions:
1. How is the arc length formula derived for polar coordinates?
2. What are the applications of cardioids in physics or engineering?
3. How would the formula change for the perimeter if the curve was in Cartesian coordinates instead of polar?
4. What is the difference between Archimedean spirals and logarithmic spirals?
5. How do we interpret the geometric meaning of "bisecting" the arc of a curve?

### Tip:
When working with polar curves, it is helpful to sketch the curve or plot it to gain a better geometric understanding of the problem.

8) Show that the arc of the upper half of the curve r = a(1 - cos θ), 0 ≤ θ ≤ π, is bisected at θ = 2π/3. 9) Determine the perimeter of the curve r = a sin³(θ/3), 0 ≤ θ ≤ 3π. 10) Find the length of the spiral r = e^(αθ) from the pole to the point (r, θ).

Learn how to solve three advanced problems in polar coordinates: bisecting the arc of a cardioid, calculating the perimeter of a sinusoidal curve, and finding the length of a spiral. These problems involve core mathematical concepts, including arc length and perimeter formulas for polar curves.

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