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The shape below is formed from two semicircles and a straight line - OCR - GCSE Maths - Question 13 - 2018 - Paper 6

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The shape below is formed from two semicircles and a straight line. The radius of the large semicircle is 8 cm. The radius of the small semicircle is 1 cm. Find an... show full transcript

Worked Solution & Example Answer:The shape below is formed from two semicircles and a straight line - OCR - GCSE Maths - Question 13 - 2018 - Paper 6

Step 1

Find the circumference of the large semicircle

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Answer

The circumference of a full circle is given by the formula C=2πrC = 2 \pi r. For the large semicircle, with a radius of 8 cm, the height is half the circumference:

Clargesemicircle=12×2π×8=8π cmC_{large semicircle} = \frac{1}{2} \times 2 \pi \times 8 = 8\pi \text{ cm}

Step 2

Find the circumference of the small semicircle

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Answer

Similarly, for the small semicircle with a radius of 1 cm:

Csmallsemicircle=12×2π×1=πextcmC_{small semicircle} = \frac{1}{2} \times 2 \pi \times 1 = \pi ext{ cm}

Step 3

Combine the lengths and add the straight line

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Answer

The total perimeter (P) of the shaded shape includes the curved parts of both semicircles and the straight line segment, which has a length given as t:

P=Clargesemicircle+Csmallsemicircle+tP = C_{large semicircle} + C_{small semicircle} + t

Now substituting the values:

P=8π+π+t=9π+tP = 8\pi + \pi + t = 9\pi + t

Thus, the expression for the exact perimeter in terms of t is:

P=9π+tP = 9\pi + t

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