Can you help me find the length of this shape and tell me how can i solve it? Please provide the answer as well so I know if I understand how to solve it.
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Here's how I would solve it. But first you need to specify what you mean by length, because there are two possible lengths: inner and outer. I also assume that the bottom loop of the S has the same inner radius as the top loop (12.5 units).

By inner, I mean in relation to the top loop of the S. This becomes the outer when traced to the lower loop.

The shape is composed of these components:

  • two semicircles: inner semicircle circumference=25π/2, outer circle (25+2×5)π=35π/2 units. The lengths of these semicircles cannot simply be combined (see later).
  • 4 arcs: inner: 3 arcs subtend an angle of 45° (⅛ of a circumference=25π/8 units each) total length=75π/8 units; 1 arc subtends an angle of 30° (1/12 of a circumference), length=25π/12 units; outer: 3 arcs=3×35π/8=105π/8; 1 arc=35π/12.
  • one rectangle: inner and outer length=30, width=5 units.

In calculating the total inner and outer lengths note that the inner length in the top part of the S becomes the outer length of the lower part of the S (and vice versa), so the total length is a combination of the inner and outer measurements.

The two measurements are:

75π/4+30+175π/8+35π/12=166.79 units approx;


105π/4+30+125π/8+25π/12=168.10 units approx.

This makes the mean length 167.45 units approx.

Another estimate of length is to lay out all components on a straight line as a long rectangle. Remember that the arcs when straightened form trapezoids, not rectangles. To calculate the length, the area of the figure must first be calculated, then divide by 5 (the width of the rectangle) to get the length.

I hope this helps.

وإليك كيف سأحلها. لكن عليك أولاً أن تحدد ما تقصده بالطول، لأن هناك طولين محتملين: الطول الداخلي والخارجي. أفترض أيضًا أن الحلقة السفلية من S لها نفس نصف القطر الداخلي للحلقة العلوية (12.5 وحدة).

بالداخلية، أعني فيما يتعلق بالحلقة العلوية للحرف S. وهذا يصبح الخارجي عند تتبعه إلى الحلقة السفلية.

يتكون الشكل من هذه المكونات:

دائرتان نصف دائريتين: محيط نصف الدائرة الداخلي = 25π/2، الدائرة الخارجية (25+2×5)π=35π/2 وحدة. لا يمكن ببساطة دمج أطوال هذه الدوائر نصف الدائرية (انظر لاحقًا).
4 أقواس: داخلي: 3 أقواس تقابل زاوية قدرها 45 درجة (⅛ المحيط=25π/8 وحدات لكل منهما) الطول الإجمالي=75π/8 وحدات؛ 1 قوس يقابل زاوية 30° (1/12 من المحيط)، الطول=25π/12 وحدة؛ الخارجي: 3 أقواس=3×35π/8=105π/8؛ 1 قوس=35π/12.
مستطيل واحد: الطول الداخلي والخارجي=30، العرض=5 وحدات.
عند حساب إجمالي الأطوال الداخلية والخارجية لاحظ أن الطول الداخلي في الجزء العلوي من S يصبح الطول الخارجي للجزء السفلي من S (والعكس صحيح)، وبالتالي فإن الطول الإجمالي هو مزيج من القياسات الداخلية والخارجية .

والقياسان هما:

75π/4+30+175π/8+35π/12=166.79 وحدة تقريبًا؛


105π/4+30+125π/8+25π/12=168.10 وحدة تقريبًا.

وهذا يجعل متوسط الطول 167.45 وحدة تقريبًا.

تقدير آخر للطول هو وضع جميع المكونات على خط مستقيم كمستطيل طويل. تذكر أن الأقواس عند تقويمها تشكل شبه منحرف، وليس مستطيلات. لحساب الطول، يجب أولاً حساب مساحة الشكل، ثم قسمتها على 5 (عرض المستطيل) للحصول على الطول.

آمل أن يساعد هذا.

by Top Rated User (1.1m points)

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