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Circles, SM-Bank 053

Town A is 135\(^\circ\) east of Town B along the equator, as shown on the diagram below.

Given the earths' radius is approximately 6400 kilometres, calculate the distance \(d\), between the two towns. Give your answer correct to the nearest whole kilometre.  (2 marks)

NOTE:  \(\text{Arc length}=\dfrac{\theta}{360}\times 2\pi r\)

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\(15\ 080\ \text{km  (nearest whole kilometre)}\)

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\(d\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{135}{360}\times 2\pi \times 6400\)
  \(=15\ 079.6447\dots\)
  \(=15\ 080\ \text{km  (nearest whole kilometre)}\)

 
\(\therefore\ \text{Towns A and B are }15\ 080\ \text{kilometres apart.}\)

Circles, SM-Bank 052

Calculate the total perimeter of the sector below, giving your answer correct to the nearest whole number.  (2 marks)

NOTE:  \(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

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\(1128\ \text{km  (nearest whole number)}\)

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\(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

\(\text{Perimeter}\) \(=2r+\dfrac{\theta}{360}\times 2\pi r\)
  \(=2\times 300+\Bigg(\dfrac{120}{360}\times 2\pi \times 300\Bigg)\)
  \(=1128.3185\dots\)
  \(=1128\ \text{km  (nearest whole number)}\)

Circles, SM-Bank 051

Calculate the total perimeter of the sector below, giving your answer correct to one decimal place.  (2 marks)

NOTE:  \(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

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\(39.0\ \text{cm  (1 d.p.)}\)

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\(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

\(\text{Perimeter}\) \(=2r+\dfrac{\theta}{360}\times 2\pi r\)
  \(=2\times 14+\Bigg(\dfrac{45}{360}\times 2\pi \times 14\Bigg)\)
  \(=38.9955\dots\)
  \(=39.0\ \text{cm  (1 d.p.)}\)

Circles, SM-Bank 050

Calculate the total perimeter of the sector below, giving your answer correct to one decimal place.  (2 marks)

NOTE:  \(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

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\(1523.6\ \text{mm  (1 d.p.)}\)

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\(\text{Perimeter}=2\times \text{radius}+\text{arc length}\)

\(\text{Perimeter}\) \(=2r+\dfrac{\theta}{360}\times 2\pi r\)
  \(=2\times 500+\Bigg(\dfrac{60}{360}\times 2\pi \times 500\Bigg)\)
  \(=1523.5987\dots\)
  \(=1523.6\ \text{mm  (1 d.p.)}\)

Circles, SM-Bank 049

Calculate the total perimeter of the quadrant below, giving your answer correct to one decimal place.  (2 marks)

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\(105.3\ \text{km  (1 d.p.)}\)

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\(\text{Perimeter}=2\times \text{radius}+\dfrac{\text{circumference}}{4}\)

\(\text{Perimeter}\) \(=2r+\dfrac{2\pi r}{4}\)
  \(=2\times 29.5+\Bigg(\dfrac{2\pi \times 29.5}{4}\Bigg)\)
  \(=105.3384 \dots\)
  \(=105.3\ \text{km  (1 d.p.)}\)

Circles, SM-Bank 048

Calculate the total perimeter of the quandrant below, giving your answer as an exact value in terms of \(\large \pi\).  (2 marks)

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\(14+\dfrac{7\pi}{2}\ \text{m}\)

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\(\text{Perimeter}=2\times \text{radius}+\dfrac{\text{circumference}}{4}\)

\(\text{Perimeter}\) \(=2r+\dfrac{2\pi r}{4}\)
  \(=2\times 7+\Bigg(\dfrac{2\pi \times 7}{4}\Bigg)\)
  \(=14+\dfrac{7\pi}{2}\ \text{m}\)

Circles, SM-Bank 046

Calculate the total perimeter of the sector below, giving your answer as an exact value in terms of \(\large \pi\).  (2 marks)

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\(100+50\pi\ \text{cm}\)

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\(\text{Perimeter}=\text{diameter}+\dfrac{\text{circumference}}{2}\)

\(\text{Perimeter}\) \(=d+\dfrac{\pi d}{2}\)
  \(=100+\Bigg(\dfrac{\pi \times 100}{2}\Bigg)\)
  \(=100+50\pi\ \text{cm}\)

Circles, SM-Bank 047

Calculate the total perimeter of the sector below, correct to one decimal place.  (2 marks)

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\(17.2\ \text{cm  (1 d.p.)}\)

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\(\text{Perimeter}=\text{diameter}+\dfrac{\text{circumference}}{2}\)

\(\text{Perimeter}\) \(=d+\dfrac{\pi d}{2}\)
  \(=6.7+\Bigg(\dfrac{\pi \times 6.7}{2}\Bigg)\)
  \(=17.2243\dots\)
  \(=17.2\ \text{cm  (1 d.p.)}\)

Circles, SM-Bank 045

Calculate the total perimeter of the sector below, correct to one decimal place.  (2 marks)

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\(442.2\ \text{cm  (1 d.p.)}\)

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\(\text{Perimeter}=\text{diameter}+\dfrac{\text{circumference}}{2}\)

\(\text{Perimeter}\) \(=d+\dfrac{\pi d}{2}\)
  \(=172+\Bigg(\dfrac{\pi \times 172}{2}\Bigg)\)
  \(=442.1769\dots\)
  \(=442.2\ \text{cm  (1 d.p.)}\)

Circles, SM-Bank 044

Calculate the arc length of the sector below, correct to two decimal places.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

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\(270.18\ \text{m  (2 d.p.)}\)

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\(\text{Diameter}=172 \text{cm}\)

\(\therefore\ \text{Radius}=\dfrac{172}{2}=86 \text{cm}\)

\(\text{Method 1}\)

\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{180}{360}\times 2\pi \times 86\)
  \(=270.1769\dots\)
  \(=270.18\ \text{m  (2 d.p.)}\)

 

\(\text{Method 2}\)

\(l\) \(=\dfrac{1}{2}\times \text{circumference}\)
  \(=\dfrac{1}{2}\times \pi d\)
  \(=\dfrac{1}{2}\times \pi \times 172\)
  \(=270.1769\dots\)
  \(=270.18\ \text{m  (2 d.p.)}\)

Circles, SM-Bank 043

Use the arc length formula below to calculate the arc length of the sector, correct to two decimal places.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

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\(7.54\ \text{m  (2 d.p.)}\)

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\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{90}{360}\times 2\pi \times 4.8\)
  \(=7.5398\dots\)
  \(=7.54\ \text{m  (2 d.p.)}\)

Circles, SM-Bank 042

Use the arc length formula below to calculate the arc length of the sector, correct to one decimal place.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

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\(12.4\ \text{mm  (1 d.p.)}\)

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\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{45}{360}\times 2\pi \times 15.8\)
  \(=12.4092\dots\)
  \(=12.4\ \text{mm  (1 d.p.)}\)

Circles, SM-Bank 041

Use the arc length formula below to calculate the arc length of the sector, correct to the nearest whole number.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

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\(628\ \text{km  (nearest whole number)}\)

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\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{120}{360}\times 2\pi \times 300\)
  \(=628.3185\dots\)
  \(=628\ \text{km  (nearest whole number)}\)

Circles, SM-Bank 040

Use the arc length formula below to calculate the arc length of the sector, correct to one decimal place.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

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\(25.0\ \text{m  (1.d.p.)}\)

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\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{135}{360}\times 2\pi \times 10.6\)
  \(=24.9756\dots\)
  \(=25.0\ \text{m  (1.d.p.)}\)

Circles, SM-Bank 039

Use the arc length formula below to calculate the arc length of the sector, correct to one decimal place.  (2 marks)

\(l=\dfrac{\theta}{360}\times 2\pi r\)

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\(4.2\ \text{cm  (1.d.p.)}\)

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\(l\) \(=\dfrac{\theta}{360}\times 2\pi r\)
  \(=\dfrac{30}{360}\times 2\pi \times 8\)
  \(=4.1887\dots\)
  \(=4.2\ \text{cm  (1.d.p.)}\)