Optimization with sectors - Radians, Arc Length and Sector Area

CAIE P1 2009 June Q5

7 marks Standard +0.3

5 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-2_385_403_1866_872} The diagram shows a circle with centre $O$. The circle is divided into two regions, $R _ { 1 }$ and $R _ { 2 }$, by the radii $O A$ and $O B$, where angle $A O B = \theta$ radians. The perimeter of the region $R _ { 1 }$ is equal to the length of the major $\operatorname { arc } A B$.

Show that $\theta = \pi - 1$.
Given that the area of region $R _ { 1 }$ is $30 \mathrm {~cm} ^ { 2 }$, find the area of region $R _ { 2 }$, correct to 3 significant figures.

CAIE P1 2013 June Q3

5 marks Standard +0.3

3 \includegraphics[max width=\textwidth, alt={}, center]{d0074ac8-42d2-49f4-a417-4a348537bccc-2_492_682_708_733} In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius 8 cm . Angle $B O A$ is $\alpha$ radians. $O A C$ is a semicircle with diameter $O A$. The area of the semicircle $O A C$ is twice the area of the sector $O A B$.

Find $\alpha$ in terms of $\pi$.
Find the perimeter of the complete figure in terms of $\pi$.

CAIE P1 2015 June Q11

10 marks Standard +0.8

11 \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-4_519_560_260_797} In the diagram, $O A B$ is a sector of a circle with centre $O$ and radius $r$. The point $C$ on $O B$ is such that angle $A C O$ is a right angle. Angle $A O B$ is $\alpha$ radians and is such that $A C$ divides the sector into two regions of equal area.

Show that $\sin \alpha \cos \alpha = \frac { 1 } { 2 } \alpha$. It is given that the solution of the equation in part (i) is $\alpha = 0.9477$, correct to 4 decimal places.
Find the ratio perimeter of region $O A C$ : perimeter of region $A C B$, giving your answer in the form $k : 1$, where $k$ is given correct to 1 decimal place.
Find angle $A O B$ in degrees. {www.cie.org.uk} after the live examination series. }

OCR MEI C2 2011 January Q9

4 marks Standard +0.3

9 Charles has a slice of cake; its cross-section is a sector of a circle, as shown in Fig. 9. The radius is $r \mathrm {~cm}$ and the sector angle is $\frac { \pi } { 6 }$ radians. He wants to give half of the slice to Jan. He makes a cut across the sector as shown. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{aa8688bb-3608-45ab-85b7-bf84889dd189-3_420_657_497_744} \captionsetup{labelformat=empty} \caption{Fig. 9}

\end{figure} Show that when they each have half the slice, $a = r \sqrt { \frac { \pi } { 6 } }$. Section B (36 marks)

OCR MEI Paper 3 2023 June Q7

6 marks Standard +0.8

7 A wire, 10 cm long, is bent to form the perimeter of a sector of a circle, as shown in the diagram. The radius is $r \mathrm {~cm}$ and the angle at the centre is $\theta$ radians. \includegraphics[max width=\textwidth, alt={}, center]{20639e13-01cc-4d96-b694-fb3cf1828f4d-07_323_204_342_242} Determine the maximum possible area of the sector, showing that it is a maximum.

Edexcel C2 Q6

10 marks Standard +0.3

6. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{1033051d-18bf-4734-a556-4c8e1c789992-3_842_963_280_392}

\end{figure} Fig. 1 shows a gardener's design for the shape of a flower bed with perimeter $A B C D . A D$ is an arc of a circle with centre $O$ and radius $5 \mathrm {~m} . B C$ is an arc of a circle with centre $O$ and radius $7 \mathrm {~m} . O A B$ and $O D C$ are straight lines and the size of $\angle A O D$ is $\theta$ radians.

Find, in terms of $\theta$, an expression for the area of the flower bed. Given that the area of the flower bed is $15 \mathrm {~m} ^ { 2 }$,
show that $\theta = 1.25$,
calculate, in m , the perimeter of the flower bed. The gardener now decides to replace arc $A D$ with the straight line $A D$.
Find, to the nearest cm , the reduction in the perimeter of the flower bed.

AQA Paper 1 2022 June Q10

12 marks Standard +0.8

10 The diagram shows a sector of a circle $O A B$. \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-16_758_796_360_623} The point $C$ lies on $O B$ such that $A C$ is perpendicular to $O B$.
Angle $A O B$ is $\theta$ radians.
10

Given the area of the triangle $O A C$ is half the area of the sector $O A B$, show that $$\theta = \sin 2 \theta$$ 10

Use a suitable change of sign to show that a solution to the equation $$\theta = \sin 2 \theta$$ lies in the interval given by $\theta \in \left[ \frac { \pi } { 5 } , \frac { 2 \pi } { 5 } \right]$

10

The Newton-Raphson method is used to find an approximate solution to the equation

$\theta = \sin 2 \theta$

10 (c) (i) Using $\theta _ { 1 } = \frac { \pi } { 5 }$ as a first approximation for $\theta$ apply the Newton-Raphson method twice

to find the value of $\theta _ { 3 }$ Give your answer to three decimal places.
10 (c) (ii) Explain how a more accurate approximation for $\theta$ can be found using the Newton-Raphson method.
10 (c) (iii) Explain why using $\theta _ { 1 } = \frac { \pi } { 6 }$ as a first approximation in the Newton-Raphson method
[0pt] [2 marks] does not lead to a solution for $\theta$.

Pre-U Pre-U 9794/2 2019 Specimen Q5

5 marks Standard +0.3

5 \includegraphics[max width=\textwidth, alt={}, center]{48b63de9-f022-4881-a187-f08e3c7d9f1a-3_570_734_219_667} The diagram shows a sector of a circle, $O M N$. The angle $M O N$ is $2 x$ radians, the radius of the circle is $r$ and $O$ is the centre.

Find expressions, in terms of $r$ and $x$, for the area, $A$, and the perimeter, $P$, of the sector.
Given that $P = 20$, show that $A = \left( \frac { 10 } { 1 + x } \right) ^ { 2 }$.
Find $\frac { \mathrm { d } A } { \mathrm {~d} x }$, and hence find the value of $x$ for which the area of the sector is a maximum.

Pre-U Pre-U 9794/2 2020 Specimen Q5

5 marks Standard +0.3

5 \includegraphics[max width=\textwidth, alt={}, center]{8a0a6e46-99cf-4217-93ad-5ed6e9d7c4ef-3_565_730_219_669} The diagram shows a sector of a circle, $O M N$. The angle $M O N$ is $2 x$ radians, the radius of the circle is $r$ and $O$ is the centre.

Find expressions, in terms of $r$ and $x$, for the area, $A$, and the perimeter, $P$, of the sector.
Given that $P = 20$, show that $A = \left( \frac { 10 } { 1 + x } \right) ^ { 2 }$.
Find $\frac { \mathrm { d } A } { \mathrm {~d} x }$, and hence find the value of $x$ for which the area of the sector is a maximum.

AQA Paper 3 2021 June Q5

13 marks Moderate -0.8

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. Find the area of this flowerbed. [2 marks]
2. Find the cost of the edging strip required for this flowerbed. [3 marks]
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]