Q. 48 Use Theorem 9.14 to show that th... [FREE SOLUTION]

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Chapter 9: Q. 48 (page 756)

Use Theorem 9.14 to show that the circumference of the circle defined by the polar equation $r = 2 a \sin 胃$ is $2 蟺 a$ .

Short Answer

Expert verified

The circumference is found to be $2 蟺 a$ by using the formula of length of polar curves.

Step by step solution

Step 1. Given Information

The function that bounds the circle is $r = 2 a \sin 胃$ .

Step 2. Use the formula of arc length of a polar curve

The region is bounded by the curve $r = 2 a \sin 胃$ .
The region is a circle, so the boundary arcs will be $胃 = 0 to 胃 = 2 蟺$ .
However, the function traces itself twice in this domain.
So, the length of the curve will be calculated as follows:

role="math" localid="1650343133331" $\frac{1}{2} \begin{array}{rcl} 鈭� \end{array} \begin{array}{rcl} 0 2 蟺 \end{array} \begin{array}{rcl} \sqrt{{(r')}^{2} + r^{2}} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} 胃 \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} \frac{1}{2} \end{array} \begin{array}{rcl} 鈭� \end{array} \begin{array}{rcl} 0 2 蟺 \end{array} \begin{array}{rcl} \sqrt{{(2 a \cos 胃)}^{2} + {(2 a \sin 胃)}^{2}} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} 胃 \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} \frac{1}{2} \end{array} \begin{array}{rcl} 脳 \end{array} \begin{array}{rcl} 2 \end{array} \begin{array}{rcl} a \end{array} \begin{array}{rcl} 鈭� \end{array} \begin{array}{rcl} 0 2 蟺 \end{array} \begin{array}{rcl} \sqrt{1} \end{array} \begin{array}{rcl} d \end{array} \begin{array}{rcl} 胃 \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} a \end{array} \begin{array}{rcl} 脳 \end{array} \begin{array}{rcl} 2 \end{array} \begin{array}{rcl} 蟺 \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} = \end{array} \begin{array}{rcl} 2 \end{array} \begin{array}{rcl} 蟺 \end{array} \begin{array}{rcl} a \end{array}$

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91影视

Use Theorem 9.14 to show that the circumference of the circle defined by the polar equation $r = 2 a \sin 胃$ is $2 蟺 a$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Use the formula of arc length of a polar curve

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91影视

Use Theorem 9.14 to show that the circumference of the circle defined by the polar equation r=2asin胃is 2蟺a.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Use the formula of arc length of a polar curve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Use Theorem 9.14 to show that the circumference of the circle defined by the polar equation $r = 2 a \sin 胃$ is $2 蟺 a$ .