Question

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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.

Answer 1

`\sf{\qquad\qquad\huge\underline{{\sf Answer}}}`

Here we go ~

Area of circle is :

`\qquad \sf  \dashrightarrow \: a = 225\pi \: \: cm {}^(2)`

`\qquad \sf  \dashrightarrow \: \pi{r}^(2) = 225 \pi`

`\qquad \sf  \dashrightarrow \: {r}^(2) = 225`

`\qquad \sf  \dashrightarrow \:r = √(225)`

`\qquad \sf  \dashrightarrow \:r = 15`

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

`\qquad \sf  \dashrightarrow \: ( \theta)/(360) (2 \pi r)`

`\qquad \sf  \dashrightarrow \: (30)/(360) (2 \sdot\pi \sdot15)`

`\qquad \sf  \dashrightarrow \: (30 \pi)/(12)`

`\qquad \sf  \dashrightarrow \:5/2 \pi`

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

Answer 2

7.85 cm (2 d.p.)

Explanation:

Step 1: Find the radius

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

Given:

• Area = 225π cm²

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

`\implies 225 \pi= \pi r^2`

`\implies r^2=225`

`\implies r=√(225)`

`\implies r=15 \:\: \sf cm`

Step 2: Find the arc length

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

Given:

• 
  `\theta` = 30°

• r = 15 cm (from step 1)

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

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

`\implies \textsf{Arc length}=(5)/(2)\pi \:\: \sf cm`

`\implies \textsf{Arc length}=7.85 \:\: \sf cm\:\:(2\:d.p.)`