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Given a circle with are 225pi cm^2.
What is the length of an arc with a central angle of 30°? Show your work.

NEED THIS DONE ASAP Thank you Given a circle with are 225pi cm2 What is the length of an arc with a central angle of 30 Show your work class=

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[tex] \sf{\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]

Here we go ~

Area of circle is :

[tex]\qquad \sf  \dashrightarrow \: a = 225\pi \: \: cm {}^{2} [/tex]

[tex]\qquad \sf  \dashrightarrow \: \pi{r}^{2} = 225 \pi[/tex]

[tex]\qquad \sf  \dashrightarrow \: {r}^{2} = 225[/tex]

[tex]\qquad \sf  \dashrightarrow \:r = \sqrt{225} [/tex]

[tex]\qquad \sf  \dashrightarrow \:r = 15[/tex]

Now, let's find the arc of circle which makes an angle 30° at centre.

[tex]\qquad \sf  \dashrightarrow \: \dfrac{ \theta}{360} (2 \pi r)[/tex]

[tex]\qquad \sf  \dashrightarrow \: \dfrac{30}{360} (2 \sdot\pi \sdot15)[/tex]

[tex]\qquad \sf  \dashrightarrow \: \dfrac{30 \pi}{12} [/tex]

[tex]\qquad \sf  \dashrightarrow \:5/2 \pi[/tex]

So, the length of required arc is 2.5 pi or 7.85 cm

semsee45

Answer:

7.85 cm  (2 d.p.)

Step-by-step explanation:

Step 1: Find the radius

[tex]\textsf{Area of a circle}=\pi r^2 \quad \textsf{(where r is the radius)}[/tex]

Given:

  • Area = 225π cm²

Substitute the given value into the formula and solve for r:

[tex]\implies 225 \pi= \pi r^2[/tex]

[tex]\implies r^2=225[/tex]

[tex]\implies r=\sqrt{225}[/tex]

[tex]\implies r=15 \:\: \sf cm[/tex]

Step 2:  Find the arc length

[tex]\textsf{Arc length}=2 \pi r\left(\dfrac{\theta}{360^{\circ}}\right) \quad \textsf{(where r is the radius and}\:\theta\:{\textsf{is the angle in degrees)}[/tex]

Given:

  • [tex]\theta[/tex] = 30°
  • r = 15 cm  (from step 1)

Substitute the given values into the formula and solve for arc length:

[tex]\implies \textsf{Arc length}=2 \pi (15) \left(\dfrac{30^{\circ}}{360^{\circ}}\right)[/tex]

[tex]\implies \textsf{Arc length}=\dfrac{5}{2}\pi \:\: \sf cm[/tex]

[tex]\implies \textsf{Arc length}=7.85 \:\: \sf cm\:\:(2\:d.p.)[/tex]

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