Difficulty: Medium
Correct Answer: 7
Explanation:
Introduction / Context:
This problem concerns the perimeter of a semicircle and its relationship with the radius. The perimeter of a semicircle includes the curved half circumferential part and the diameter. The question provides the total perimeter and asks for the radius. This tests knowledge of circle formulas and algebraic manipulation.
Given Data / Assumptions:
Concept / Approach:
The perimeter of a full circle is 2 * pi * r. A semicircle has half this curved length plus the straight diameter. Therefore, the perimeter of a semicircle is P = pi * r + 2 * r. We set this equal to the given 36 cm and solve for r using the approximate value of pi.
Step-by-Step Solution:
Step 1: Write the formula for the perimeter of a semicircle: P = pi * r + 2 * r.Step 2: Substitute P = 36: 36 = pi * r + 2 * r.Step 3: Factor r: 36 = r * (pi + 2).Step 4: Use pi = 22 / 7. Then pi + 2 = 22 / 7 + 2 = 22 / 7 + 14 / 7 = 36 / 7.Step 5: Substitute into the equation: 36 = r * (36 / 7).Step 6: Solve for r: r = 36 * (7 / 36) = 7 cm.
Verification / Alternative check:
Substitute r = 7 cm back into the original formula for the perimeter.Compute P = pi * r + 2 * r = (22 / 7) * 7 + 2 * 7 = 22 + 14 = 36 cm.This matches the given perimeter, confirming that r = 7 cm is correct.
Why Other Options Are Wrong:
If r were 14 cm, the semicircle perimeter would be pi * 14 + 28, which is much larger than 36 cm.Values like 13 or 21 cm would similarly produce perimeters far from 36 cm when substituted into the formula.Thus, none of these values satisfy the equation 36 = pi * r + 2 * r.
Common Pitfalls:
A frequent mistake is to use only half the circumference pi * r and forget to add the diameter.Another error is to use 2 * pi * r directly as the perimeter of the semicircle, which corresponds to a full circle instead.Misusing the approximation for pi or incorrect algebra when isolating r can also lead to wrong answers.
Final Answer:
The radius of the semicircle is 7 centimetres.
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