The curve below consists of three semicircles: AB, BC, and CD. The diameter of AB is 2, the diameter of BC is twice the diameter of AB, and the diameter of CD is twice the diameter of BC. What is the total length of the curve?
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Notice a pattern in the lengths: \(\pi, 2\pi, 4\pi\). This is a geometric progression. For complex figures made of simple shapes, calculate the dimension of each shape first, then apply the appropriate formula for each, and finally sum the results.
Step 1: Understanding the Concept:
The total length of the curve is the sum of the lengths of the three semicircles. The length of a semicircle is half its circumference. Step 2: Key Formula or Approach:
The formula for the circumference of a full circle is \(C = \pi d\), where \(d\) is the diameter.
The length of a semicircle is therefore \(\frac{1}{2}\pi d\). Step 3: Detailed Explanation:
1. Determine the diameters of the three semicircles.
Diameter of AB (\(d_{AB}\)) = 2.
Diameter of BC (\(d_{BC}\)) = 2 \(\times\) \(d_{AB}\) = 2 \(\times\) 2 = 4.
Diameter of CD (\(d_{CD}\)) = 2 \(\times\) \(d_{BC}\) = 2 \(\times\) 4 = 8.
2. Calculate the length of each semicircular arc.
Length of AB = \(\frac{1}{2}\pi d_{AB} = \frac{1}{2}\pi(2) = \pi\).
Length of BC = \(\frac{1}{2}\pi d_{BC} = \frac{1}{2}\pi(4) = 2\pi\).
Length of CD = \(\frac{1}{2}\pi d_{CD} = \frac{1}{2}\pi(8) = 4\pi\).
3. Calculate the total length of the curve.
Total Length = Length of AB + Length of BC + Length of CD.
Total Length = \(\pi + 2\pi + 4\pi = 7\pi\). Step 4: Final Answer:
The total length of the curve is \(7\pi\).
