https://file.aiquickdraw.com/mathbot/file/1729890240746yxosdwx5.jpg

Determine the perimeter of the shape below as an exact value (note the shape is outlined by solid lines)

The shape consists of several segments:

1. A straight line of length \(4 \, \text{cm}\).
2. A circular arc with radius \(2 \, \text{cm}\) subtended by a \(120^\circ\) angle.
3. A straight line of length \(3 \, \text{cm}\).
4. A circular arc with radius \(3 \, \text{cm}\) subtended by a \(210^\circ\) angle.

### Step 1: Perimeter of straight lines
The total length of the straight segments is:
\[
4 \, \text{cm} + 3 \, \text{cm} = 7 \, \text{cm}.
\]

### Step 2: Perimeter contribution from the arcs
The length of an arc is given by the formula:
\[
\text{Arc length} = 2 \pi r \times \frac{\theta}{360^\circ},
\]
where \(r\) is the radius and \(\theta\) is the central angle.

#### Arc 1:
For the first arc with radius \(2 \, \text{cm}\) and angle \(120^\circ\):
\[
\text{Arc length} = 2 \pi \times 2 \times \frac{120^\circ}{360^\circ} = \frac{4\pi}{3} \, \text{cm}.
\]

#### Arc 2:
For the second arc with radius \(3 \, \text{cm}\) and angle \(210^\circ\):
\[
\text{Arc length} = 2 \pi \times 3 \times \frac{210^\circ}{360^\circ} = \frac{7\pi}{3} \, \text{cm}.
\]

### Step 3: Total perimeter
Adding the straight lines and arc lengths together:
\[
\text{Perimeter} = 7 + \frac{4\pi}{3} + \frac{7\pi}{3} = 7 + \frac{11\pi}{3} \, \text{cm}.
\]
Thus, the exact value of the perimeter is:
\[
7 + \frac{11\pi}{3} \, \text{cm}.
\]

Let me know if you need more details or have any questions! 

Here are 5 related questions for you:
1. How do you calculate the length of a circular arc?
2. What is the formula for the circumference of a full circle?
3. How does the angle affect the length of an arc?
4. Can a sector's area be calculated using a similar formula as the arc length?
5. How would the perimeter change if the angles were different?

**Tip:** For circular arcs, always double-check whether the angle is given in degrees or radians before applying the formula!

This solution provides a step-by-step approach to determine the perimeter of a composite shape consisting of straight lines and circular arcs. The key concepts include calculating the arc length using the formula 2πr × (θ/360°) and summing the perimeter contributions from straight lines and arcs. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation