1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta

\includegraphics{figure_6} The diagram shows triangle \(ABC\) in which \(AC = 8\) cm and \(\angle BAC = \angle BCA = 30°\).

1. Find the area of triangle \(ABC\) in the form \(k\sqrt{3}\). [4]

The point \(M\) is the mid-point of \(AC\) and the points \(N\) and \(O\) lie on \(AB\) and \(BC\) such that \(MN\) and \(MO\) are arcs of circles with centres \(A\) and \(C\) respectively.

1. Show that the area of the shaded region \(BNMO\) is \(\frac{8}{3}(2\sqrt{3} - \pi)\) cm\(^2\). [4]

Fig. 11.1 shows a village green which is bordered by 3 straight roads AB, BC and CA. The road AC runs due North and the measurements shown are in metres. \includegraphics{figure_1}

1. Calculate the bearing of B from C, giving your answer to the nearest 0.1°. [4]
2. Calculate the area of the village green. [2]

The road AB is replaced by a new road, as shown in Fig. 11.2. The village green is extended up to the new road. \includegraphics{figure_2} The new road is an arc of a circle with centre O and radius 130 m.

1. (A) Show that angle AOB is 1.63 radians, correct to 3 significant figures. [2] (B) Show that the area of land added to the village green is 5300 m² correct to 2 significant figures. [4]

\emph{Arrowline Enterprises} is considering two possible logos: \includegraphics{figure_6}

1. Fig. 10.1 shows the first logo ABCD. It is symmetrical about AC. Find the length of AB and hence find the area of this logo. [4]
2. Fig. 10.2 shows a circle with centre O and radius 12.6 cm. ST and RT are tangents to the circle and angle SOR is 1.82 radians. The shaded region shows the second logo. Show that ST = 16.2 cm to 3 significant figures. Find the area and perimeter of this logo. [8]

Archimedes, about 2200 years ago, used regular polygons inside and outside circles to obtain approximations for \(\pi\).

1. Fig. 8.1 shows a regular 12-sided polygon inscribed in a circle of radius 1 unit, centre O. AB is one of the sides of the polygon. C is the midpoint of AB. Archimedes used the fact that the circumference of the circle is greater than the perimeter of this polygon. \includegraphics{figure_1}
   1. Show that \(\text{AB} = 2 \sin 15°\). [2]
   2. Use a double angle formula to express \(\cos 30°\) in terms of \(\sin 15°\). Using the exact value of \(\cos 30°\), show that \(\sin 15° = \frac{1}{2}\sqrt{2 - \sqrt{3}}\). [4]
   3. Use this result to find an exact expression for the perimeter of the polygon. Hence show that \(\pi > 6\sqrt{2 - \sqrt{3}}\). [2]
2. In Fig. 8.2, a regular 12-sided polygon lies outside the circle of radius 1 unit, which touches each side of the polygon. F is the midpoint of DE. Archimedes used the fact that the circumference of the circle is less than the perimeter of this polygon. \includegraphics{figure_2}
   1. Show that \(\text{DE} = 2 \tan 15°\). [2]
   2. Let \(t = \tan 15°\). Use a double angle formula to express \(\tan 30°\) in terms of \(t\). Hence show that \(t^2 + 2\sqrt{3}t - 1 = 0\). [3]
   3. Solve this equation, and hence show that \(\pi < 12(2 - \sqrt{3})\). [4]
3. Use the results in parts (i)(C) and (ii)(C) to establish upper and lower bounds for the value of \(\pi\), giving your answers in decimal form. [2]

The diagram below shows a sector of a circle. \includegraphics{figure_3} The radius of the circle is 4cm and \(\theta = 0.8\) radians. Find the area of the sector. Circle your answer. [1 mark] $$1.28\text{cm}^2 \quad 3.2\text{cm}^2 \quad 6.4\text{cm}^2 \quad 12.8\text{cm}^2$$

The diagram shows a sector \(AOB\) of a circle with centre \(O\) and radius \(r\) cm. \includegraphics{figure_5} The angle \(AOB\) is \(\theta\) radians The sector has area 9 cm\(^2\) and perimeter 15 cm.

1. Show that \(r\) satisfies the equation \(2r^2 - 15r + 18 = 0\) [4 marks]
2. Find the value of \(\theta\). Explain why it is the only possible value. [4 marks]

A ball is projected forward from a fixed point, \(P\), on a horizontal surface with an initial speed \(u\text{ ms}^{-1}\), at an acute angle \(\theta\) above the horizontal. The ball needs to first land at a point at least \(d\) metres away from \(P\). You may assume the ball may be modelled as a particle and that air resistance may be ignored. Show that $$\sin 2\theta \geq \frac{dg}{u^2}$$ [6 marks]

A circle has equation \(x^2 + y^2 - 6x - 8y = 264\) \(AB\) is a chord of the circle. The angle at the centre of the circle, subtended by \(AB\), is \(0.9\) radians, as shown in the diagram below. \includegraphics{figure_5} Find the area of the minor segment shaded on the diagram. Give your answer to three significant figures. [5 marks]

A gardener is creating flowerbeds in the shape of sectors of circles. The gardener uses an edging strip around the perimeter of each of the flowerbeds. The cost of the edging strip is £1.80 per metre and can be purchased for any length. One of the flowerbeds has a radius of 5 metres and an angle at the centre of 0.7 radians as shown in the diagram below. \includegraphics{figure_5}

1. 1. Find the area of this flowerbed. [2 marks]
   2. Find the cost of the edging strip required for this flowerbed. [3 marks]
2. A flowerbed is to be made with an area of 20 m²
   1. Show that the cost, £\(C\), of the edging strip required for this flowerbed is given by $$C = \frac{18}{5}\left(\frac{20}{r} + r\right)$$ where \(r\) is the radius measured in metres. [3 marks]
   2. Hence, show that the minimum cost of the edging strip for this flowerbed occurs when \(r \approx 4.5\) Fully justify your answer. [5 marks]

A new design for a company logo is to be made from two sectors of a circle, \(ORP\) and \(OQS\), and a rhombus \(OSTR\), as shown in the diagram below. \includegraphics{figure_7} The points \(P\), \(O\) and \(Q\) lie on a straight line and the angle \(ROS\) is \(\theta\) radians. A large copy of the logo, with \(PQ = 5\) metres, is to be put on a wall.

1. Show that the area of the logo, \(A\) square metres, is given by $$A = \frac{25}{8}(\pi - \theta + 2\sin\theta)$$ [4 marks]
2. 1. Show that the maximum value of \(A\) occurs when \(\theta = \frac{\pi}{3}\) Fully justify your answer. [6 marks]
   2. Find the exact maximum value of \(A\) [2 marks]
3. Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if \(PQ\) were increased to 10 metres. [2 marks]

The diagram below shows a sector of a circle \(OAB\). The chord \(AB\) divides the sector into a triangle and a shaded segment. Angle \(AOB\) is \(\frac{\pi}{6}\) radians. The radius of the sector is 18 cm. \includegraphics{figure_5} Show that the area of the shaded segment is $$k(\pi - 3) \text{cm}^2$$ where \(k\) is an integer to be found. [3 marks]

A wooden frame is to be made to support some garden decking. The frame is to be in the shape of a sector of a circle. The sector \(OAB\) is shown in the diagram, with a wooden plank \(AC\) added to the frame for strength. \(OA\) makes an angle of \(\theta\) with \(OB\). \includegraphics{figure_2}

1. Show that the exact value of \(\sin\theta\) is \(\frac{4\sqrt{14}}{15}\) [3 marks]
2. Write down the value of \(\theta\) in radians to 3 significant figures. [1 mark]
3. Find the area of the garden that will be covered by the decking. [2 marks]

The diagram below shows a circle centre O, radius 4 cm. Points A and B lie on the circumference such that arc AB is 5 cm. \includegraphics{figure_2}

1. Calculate the angle subtended at O by the arc AB. [2]
2. Determine the area of the sector OAB. [2]

The diagram below shows a badge \(ODC\). The shape \(OAB\) is a sector of a circle centre \(O\) and radius \(r\) cm. The shape \(ODC\) is a sector of a circle with the same centre \(O\). The length \(AD\) is \(5\) cm and angle \(AOB\) is \(\frac{\pi}{5}\) radians. The area of the shaded region, \(ABCD\), is \(\frac{13\pi}{2}\) cm\(^2\). \includegraphics{figure_3}

1. Determine the value of \(r\). [4]
2. Calculate the perimeter of the shaded region. [3]

\includegraphics{figure_1} Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector \(BCDF\), of a circle centre \(F\), joined to two congruent triangles \(ABF\) and \(EDF\). Given that - \(AFE\) is a straight line - \(AF = FE = 10.7\)m - \(BF = FD = 9.2\)m - angle \(BFD = 1.82\) radians find

1. the perimeter of the stage, in metres, to one decimal place, [5]
2. the area of the stage, in m², to one decimal place. [4]

\includegraphics{figure_1} The shape \(ABCDOA\), as shown in Figure 1, consists of a sector \(COD\) of a circle centre \(O\) joined to a sector \(AOB\) of a different circle, also centre \(O\). Given that arc length \(CD = 3\) cm, \(\angle COD = 0.4\) radians and \(AOD\) is a straight line of length 12 cm,

1. find the length of \(OD\), [2]
2. find the area of the shaded sector \(AOB\). [3]

The diagram shows a triangle \(ABC\), and a sector \(ACD\) of a circle with centre \(A\). It is given that \(AB = 11\) cm, \(BC = 8\) cm, angle \(ABC = 0.8\) radians and angle \(DAC = 1.7\) radians. The shaded segment is bounded by the line \(DC\) and the arc \(DC\). \includegraphics{figure_7}

1. Show that the length of \(AC\) is \(7.90\) cm, correct to 3 significant figures. [3]
2. Find the area of the shaded segment. [3]
3. Find the perimeter of the shaded segment. [4]

\includegraphics{figure_1} Figure 1 shows the plan view of a design for a stage at a concert. The stage is modelled as a sector \(BCDF\), of a circle centre \(F\), joined to two congruent triangles \(ABF\) and \(EDF\). Given that \(AFE\) is a straight line, \(AF = FE = 10.7\) m, \(BF = FD = 9.2\) m and angle \(BFD = 1.82\) radians, find the perimeter of the stage, in metres, to one decimal place. [4]

\includegraphics{figure_2} The shape \(AOCBA\), shown in Figure 2, consists of a sector \(AOB\) of a circle centre \(O\) joined to a triangle \(BOC\). The points \(A\), \(O\) and \(C\) lie on a straight line with \(AO = 7.5\) cm and \(OC = 8.5\) cm. The size of angle \(AOB\) is 1.2 radians. Find, in cm, the perimeter of the shape \(AOCBA\), giving your answer to one decimal place. [5]

\includegraphics{figure_7} The figure above shows a circular sector \(OAB\) whose centre is at \(O\). The radius of the sector is 60 cm. The points \(C\) and \(D\) lie on \(OA\) and \(OB\) respectively, so that \(|OC| = |OD| = 24\) cm. Given that the length of the arc \(AB\) is 48 cm, find the area of the shaded region \(ABDC\), correct to the nearest cm\(^2\). [5 marks]

\includegraphics{figure_10} The figure above shows solid right prism of height \(h\) cm. The cross section of the prism is a circular sector of radius \(r\) cm, subtending an angle of 2 radians at the centre.

1. Given that the volume of the prism is 1000 cm\(^3\), show clearly that $$S = 2r^2 + \frac{4000}{r},$$ where \(S\) cm\(^2\) is the total surface area of the prism. [5]
2. Hence determine the value of \(r\) and the value of \(h\) which make \(S\) least, fully justifying your answer. [7]

\includegraphics{figure_3} The diagram shows a sector \(AOB\) of a circle with centre \(O\). The length of the arc \(AB\) is \(6\) cm and the area of the sector \(AOB\) is \(24\) cm\(^2\). Find the area of the shaded segment enclosed by the arc \(AB\) and the chord \(AB\), giving your answer correct to \(3\) significant figures. [6]

The diagram shows a sector \(AOB\) of a circle with centre \(O\) and radius \(r\) cm. \includegraphics{figure_4} The angle \(AOB\) is \(\theta\) radians. The arc length \(AB\) is 15 cm and the area of the sector is 45 cm\(^2\).

1. Find the values of \(r\) and \(\theta\). [4]
2. Find the area of the segment bounded by the arc \(AB\) and the chord \(AB\). [3]