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Find the circumference of the circle. Then, find the length of each bolded arc. Use appropriate notation

Find the circumference of the circle. Then, find the length of each bolded arc. Use-example-1
User JadedEric
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Answer:


\text{1) }\\\text{Circumference: }24\pi \text{ m}},\\\text{Length of bolded arc: }18\pi \text{ m}\\\\\text{3)}\\\text{Circumference. }4\pi \text{ mi},\\\text{Length of bolded arc: } (3\pi)/(2)\text{ mi}

Explanation:

The circumference of a circle with radius
r is given by
C=2\pi r. The length of an arc is makes up part of this circumference, and is directly proportion to the central angle of the arc. Since there are 360 degrees in a circle, the length of an arc with central angle
\theta^(\circ) is equal to
2\pi r\cdot (\theta)/(360).

Formulas at a glance:

  • Circumference of a circle with radius
    r:
    C=2\pi r
  • Length of an arc with central angle
    \theta^(\circ):
    \ell_(arc)=2\pi r\cdot (\theta)/(360)

Question 1:

The radius of the circle is 12 m. Therefore, the circumference is:


C=2\pi r,\\C=2(\pi)(12)=\boxed{24\pi\text{ m}}

The measure of the central angle of the bolded arc is 270 degrees. Therefore, the measure of the bolded arc is equal to:


\ell_(arc)=24\pi \cdot (270)/(360),\\\\\ell_(arc)=24\pi \cdot (3)/(4),\\\\\ell_(arc)=\boxed{18\pi\text{ m}}

Question 2:

In the circle shown, the radius is marked as 2 miles. Substituting
r=2 into our circumference formula, we get:


C=2(\pi)(2),\\C=\boxed{4\pi\text{ mi}}

The measure of the central angle of the bolded arc is 135 degrees. Its length must then be:


\ell_(arc)=4\pi \cdot (135)/(360),\\\ell_(arc)=1.5\pi=\boxed{(3\pi)/(2)\text{ mi}}

User Aras
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