Arc length and sector area

The diagram shows a sector AOB of a circle with centre O and radius 8 cm.

The angle AOB is 1.4 radians.

Find the area of the sector.

A circle with centre C has equation

The graph with equation

y = │ 3 x – 36 │ – 8

intersects the circle at the points A, B and D as shown in the diagram.

Point D is vertically below point C

State the coordinates of C

State the coordinates of D

The coordinates of A are (a,10)

Find the value of a

Fully justify your answer.

Find, in radians, the angle ADB

Give your answer to three significant figures.

Hence or otherwise find the length of the minor arc AB

Give your answer to three significant figures.

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.

The points P, O and Q lie on a straight line and the angle ROS is radians.

A large copy of the logo, with PQ 5 metres, is to be put on a wall.

Show that the area of the logo, A square metres, is given by

Show that the maximum value of A occurs when

Fully justify your answer.

Find the exact maximum value of A

Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if PQ were increased to 10 metres.

The diagram shows a sector of a circle OAB.

The point C lies on OB such that AC is perpendicular to OB.

Angle AOB is radians.

Given the area of the triangle OAC is half the area of the sector OAB, show that

Use a suitable change of sign to show that a solution to the equation

lies in the interval given by

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

Using ₁ as a first approximation for apply the Newton-Raphson method twice

Give your answer to three decimal places.

Explain how a more accurate approximation for can be found using the Newton-Raphson method.

Explain why using ₁ as a first approximation in the Newton-Raphson method does not lead to a solution for .

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.

Find the area of this flowerbed.

Find the cost of the edging strip required for this flowerbed.

A flowerbed is to be made with an area of 20 m²

Show that the cost, £ C, of the edging strip required for this flowerbed is given by

where r is the radius measured in metres.

Hence, show that the minimum cost of the edging strip for this flowerbed occurs

Fully justify your answer.

The diagram below shows a sector of a circle.

The radius of the circle is 4 cm and 08 radians.

Find the area of the sector.

Circle your answer.

2 2 2 2 1.28 cm 3.2 cm 6.4 cm 12.8 cm

A circle has equation x² y² 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.

Find the area of the minor segment shaded on the diagram.

Give your answer to three significant figures.

The diagram shows a sector of a circle OAB.

C is the midpoint of OB.

Angle AOB is radians.

Given that the area of the triangle OAC is equal to one quarter of the area of the sector OAB, show that 2 sin

Use the Newton-Raphson method with ₁ , to find ₃ as an approximation for . Give your answer correct to five decimal places.

Given that 1.89549 to five decimal places, find an estimate for the percentage error in the approximation found in part (b).