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Math  /  Geometry

Question4. Find the length of the arc of a circle of diameter 14 meters subtended by a central angle of 5π6\frac{5 \pi}{6} radians (2 marks)

Studdy Solution

STEP 1

What is this asking? We need to find how long the curved part of a circle is, given its size and the angle it sweeps out. Watch out! Don't mix up radius and diameter!
Also, make sure your angle is in radians, not degrees.

STEP 2

1. Find the radius
2. Calculate the arc length

STEP 3

We're given the **diameter**, which is $14\$14 meters.
The **radius** is *half* the diameter.
Remember, the radius is the distance from the center of the circle to any point on the edge.
It's like the arm of the circle reaching out!

STEP 4

So, to get the **radius**, we divide the **diameter** by 22:
radius=diameter2=142=7 meters.\text{radius} = \frac{\text{diameter}}{2} = \frac{14}{2} = 7 \text{ meters}.

STEP 5

Our **radius** is $7\$7 meters.
Keep this number handy, we'll need it soon!

STEP 6

The formula for **arc length** (*s*) is:
s=r⋅θs = r \cdot \thetawhere *r* is the **radius** and *θ* (*theta*) is the **central angle** in radians.
This formula is like magic!
It connects the **size of the circle** (radius) with the **angle it sweeps** (theta) to tell us how long the **curved part** is (arc length).

STEP 7

We know our **radius** is 77 meters and our **central angle** is 5π6\frac{5\pi}{6} radians.
Let's plug those values into our formula:
s=7⋅5π6s = 7 \cdot \frac{5\pi}{6}

STEP 8

Now, we multiply:
s=7⋅5π6=35π6 meters.s = \frac{7 \cdot 5\pi}{6} = \frac{35\pi}{6} \text{ meters}.

STEP 9

The length of the arc is 35π6\frac{35\pi}{6} meters.

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