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Consider circle [tex]\( C \)[/tex] with a radius of 5 cm and a central angle measure of [tex]\( 60^{\circ} \)[/tex].

1. What fraction of the whole circle is arc RS?

2. What is the approximate circumference of the circle? (cm)

3. What is the approximate length of arc RS? (cm)

Sagot :

GinnyAnswer
To solve these problems, we need to break down each one step by step:

### 1. Fraction of the whole circle that arc RS represents
Given:
- The central angle measures [tex]\(60^\circ\)[/tex].

The total angle of a circle is [tex]\(360^\circ\)[/tex]. The fraction of the circle that arc RS represents can be found by dividing the central angle by the total angle of the circle:

[tex]\[ \text{Fraction of the circle} = \frac{\text{Central angle}}{360^\circ} = \frac{60^\circ}{360^\circ} \][/tex]

Simplifying this fraction:

[tex]\[ \frac{60}{360} = \frac{1}{6} \approx 0.16666666666666666 \][/tex]

So, the fraction of the whole circle that arc RS represents is approximately [tex]\(0.1667\)[/tex].

### 2. Approximate circumference of the circle
Given:
- The radius of the circle is [tex]\(5\)[/tex] cm.

The formula for the circumference of a circle is:

[tex]\[ \text{Circumference} = 2 \pi \times \text{radius} \][/tex]

Substituting the given radius:

[tex]\[ \text{Circumference} = 2 \pi \times 5 \][/tex]

Using the approximate value of [tex]\(\pi \approx 3.1416\)[/tex], we get:

[tex]\[ \text{Circumference} \approx 2 \times 3.1416 \times 5 = 31.41592653589793 \text{ cm} \][/tex]

So, the approximate circumference of the circle is [tex]\(31.4159\)[/tex] cm.

### 3. Approximate length of arc RS
To find the length of arc RS, we use the fraction of the circle that the arc represents (found in step 1) and the circumference of the circle (found in step 2).

The formula for the length of an arc is:

[tex]\[ \text{Length of the arc} = \left(\text{Fraction of the circle}\right) \times \text{Circumference} \][/tex]

Substituting the known values:

[tex]\[ \text{Length of arc RS} = \left(0.16666666666666666\right) \times 31.41592653589793 \approx 5.235987755982988 \text{ cm} \][/tex]

So, the approximate length of arc RS is [tex]\(5.2360\)[/tex] cm.

### Summary
- The fraction of the whole circle that arc RS represents is approximately [tex]\(0.16666666666666666\)[/tex].
- The approximate circumference of the circle is [tex]\(31.41592653589793\)[/tex] cm.
- The approximate length of arc RS is [tex]\(5.235987755982988\)[/tex] cm.
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