To find the exact perimeter of the figure consisting of a semicircle and a rectangle, we need to determine the dimensions and calculate the contributions from both shapes.
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Rectangle Dimensions:
- Let’s denote the width of the rectangle as \( w \) and the height as \( h \).
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Semicircle Details:
- Assuming the semicircle is sitting on one of the shorter sides of the rectangle, its diameter will be the same as the width \( w \).
- Therefore, the radius \( r \) of the semicircle is \( \frac{w}{2} \).
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Perimeter Calculation:
- The perimeter of the rectangle includes two heights and the width across the top (which is covered by the semicircle).
- The contribution to the perimeter from the rectangle is \( 2h + w \).
- The semicircle adds half the circumference to the perimeter, calculated as:
\[ \text{Circumference of full circle} = 2\pi r = 2\pi \left(\frac{w}{2}\right) = \pi w \]
So, the contribution from the semicircle is \( \frac{1}{2} \times \text{Circumference} = \frac{1}{2} \times \pi w = \frac{\pi w}{2} \).
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Total Perimeter:
- Therefore, the total perimeter \( P \) is:
\[ P = 2h + w + \frac{\pi w}{2} \]
Now, we need specific values for \( w \) and \( h \) to simplify it further and match it against the provided options. Since these values are not provided, we can't compute a precise answer. However, examining the answer choices suggests that simplifications may lead to a form included in them.
Assuming \( w \) and \( h \) allow for matching:
For example, if \( h = 6 \) and \( w = 6 \):
\[ P = 2(6) + 6 + \frac{\pi(6)}{2} = 12 + 6 + 3\pi = 18 + 3\pi \Rightarrow (3\pi + 18) \text{ mm} \]
If the values are:
- \( h = 6 \) and
- \( w = 12 \):
\[ P = 12 + 12 + \frac{1}{2} \times 2\pi(12) = 12 + 12 + 6\pi = 24 + 6 \pi \Rightarrow (6\pi + 12) \text{ mm} \]
To finalize which answer corresponds, more details regarding relationships between \( w \) and \( h \) are needed, but if we assume:
- Height corresponding alignments,
- Diameter relations confirm shapes,
We'd ultimately conclude on dimensional evaluations.
In summary, if none of the initial conditions were met, assume the actual option aligning closest.
Thus, the exact perimeter based on various setups would lead either toward: (3π + 18) mm, (6π + 12) mm, or others looking from practical rectangles.
If the overall assessment suggests confirming when provided further specs. Adding those enables responses correct paths toward exact options from available calculations.