Question

the curved parts of the figure are arcs centered at points a and c. what is the approximate length of boundary abcd? use the value π = 3.14, and round the answer to one decimal place.
image of the figure with points d, c (5 units apart), 30° angle at c, point a, 120° angle at a, point b (5 units from a), and multiple-choice options: a. 17.8, b. 23.6, c. 23.1, d. 28.1

Explanation:

Step1: Analyze the boundary components

The boundary \(ABCD\) consists of two line segments (\(DC\) and \(AB\)) and two arcs (arc \(AD\) centered at \(A\) and arc \(BC\) centered at \(C\)).

Step2: Calculate the length of line segments

• \(DC = 5\) (given)
• \(AB = 5\) (given)

Total length of line segments: \(5 + 5 = 10\)

Step3: Calculate the length of arc \(AD\) (centered at \(A\))

The central angle for arc \(AD\) is \(30^\circ\) (since the angle at \(C\) is \(30^\circ\) and the triangles/symmetry imply this). The radius \(r = 5\).
The formula for the length of an arc is \(L = \frac{\theta}{360^\circ} \times 2\pi r\)
For arc \(AD\): \(\theta = 30^\circ\), \(r = 5\)
\(L_{AD} = \frac{30^\circ}{360^\circ} \times 2 \times 3.14 \times 5\)
\(L_{AD} = \frac{1}{12} \times 31.4 \approx 2.617\)

Step4: Calculate the length of arc \(BC\) (centered at \(C\))

The central angle for arc \(BC\) is \(120^\circ\) (given). The radius \(r = 5\).
Using the arc length formula: \(L = \frac{\theta}{360^\circ} \times 2\pi r\)
For arc \(BC\): \(\theta = 120^\circ\), \(r = 5\)
\(L_{BC} = \frac{120^\circ}{360^\circ} \times 2 \times 3.14 \times 5\)
\(L_{BC} = \frac{1}{3} \times 31.4 \approx 10.467\)

Step5: Calculate the total length of the boundary

Total length \(= DC + AB + L_{AD} + L_{BC}\)
\(= 5 + 5 + 2.617 + 10.467\)
\(= 23.084 \approx 23.1\) (rounded to one decimal place)

Answer:

C. 23.1