Question

Question was provided via image upload.

Explanation:

Identify the given values and geometric components

The figure consists of two curved sectors sharing a boundary.

• The curved parts of the figure are circular arcs centered at points \(A\) and \(C\).
• For the arc \(DE\) centered at \(C\):
• The radius \(CD = 5\).
• The central angle \(\angle DCE = 30^\circ\).
• For the arc \(BE\) centered at \(A\):
• The radius \(AB = 5\).
• The central angle \(\angle BAE = 120^\circ\).
• We need to find the approximate perimeter of the shaded region.
• The boundary of the shaded region consists of:

1. Straight segment \(AB\) of length \(5\).
2. Circular arc \(BE\) centered at \(A\) with radius \(5\) and central angle \(120^\circ\).
3. Straight segment \(CD\) of length \(5\).
4. Circular arc \(DE\) centered at \(C\) with radius \(5\) and central angle \(30^\circ\).

• Use \(\pi \approx 3.14\) and round the final answer to one decimal place.

Calculate the length of arc BE

The formula for the arc length of a sector is:

$$ L = \frac{\theta}{360^\circ} \times 2\pi r $$

For arc \(BE\):

• Radius \(r_1 = 5\)
• Angle \(\theta_1 = 120^\circ\)

$$ L_{BE} = \frac{120}{360} \times 2 \times 3.14 \times 5 = \frac{1}{3} \times 31.4 \approx 10.47 $$

Calculate the length of arc DE

For arc \(DE\):

• Radius \(r_2 = 5\)
• Angle \(\theta_2 = 30^\circ\)

$$ L_{DE} = \frac{30}{360} \times 2 \times 3.14 \times 5 = \frac{1}{12} \times 31.4 \approx 2.62 $$

Sum the boundary lengths to find the perimeter

The total perimeter \(P\) is the sum of the two straight segments and the two arcs:

$$ P = AB + L_{BE} + CD + L_{DE} $$

Substitute the values:

$$ P = 5 + 10.47 + 5 + 2.62 = 23.09 $$

Rounding to one decimal place gives \(23.1\).

Match with the multiple-choice options

Let's look at the visible options in the image:

• Option 1: \(17.3\)
• Option 2: \(23.0\)
• Option 3: \(23.1\)
• Option 4: \(28.1\)

Our calculated value of \(23.1\) matches the third option.

Answer:

• (A) 17.3
• (B) 23.0
• (C) 23.1 (Correct answer)
• (D) 28.1