Polar coordinates

The curve C shown in Figure 1 has polar equation

where a is a constant.

The tangents to C at the points A and B are perpendicular to the initial line.

Use calculus to determine the polar coordinates of A and B

The curve C models the perimeter of the surface of a swimming pool.

Given that, according to the model, the distance across the pool from A to B is 10 m,

Use algebraic integration to determine the surface area of the swimming pool, according to the model.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows the design for a bathing pool.

The pool, P, shown unshaded in Figure 1, is surrounded by a tiled area, T, shown shaded in Figure 1.

The tiled area is bounded by the edge of the pool and by a circle, C, with radius 6 m.

The centre of the pool and the centre of the circle are the same point.

The edge of the pool is modelled by the curve with polar equation

where a is a positive constant.

Given that the shortest distance between the edge of the pool and the circle C is 0.5 m,

determine the value of a.

Hence, using algebraic integration, determine, according to the model, the exact area of T.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows the design for a bathing pool.

The pool, P, shown unshaded in Figure 1, is surrounded by a tiled area, T, shown shaded in Figure 1.

The tiled area is bounded by the edge of the pool and by a circle, C, with radius 6 m.

The centre of the pool and the centre of the circle are the same point.

The edge of the pool is modelled by the curve with polar equation

where a is a positive constant.

Given that the shortest distance between the edge of the pool and the circle C is 0.5 m,

determine the value of a.

Hence, using algebraic integration, determine, according to the model, the exact area of T.

Figure 1 shows a sketch of the curve C with equation

Figure 1 also shows the tangent to C at the point A. This tangent is perpendicular to the initial line.

The finite region R, shown shaded in Figure 1, is bounded by C, the tangent at A and the initial line.

Use calculus to show that the exact area of R is (1 – ln 2)

Figure 1 shows a sketch of the curve C with equation

Figure 1 also shows the tangent to C at the point A. This tangent is perpendicular to the initial line.

The finite region R, shown shaded in Figure 1, is bounded by C, the tangent at A and the initial line.

Use calculus to show that the exact area of R is (1 – ln 2)

The curve C has equation

where a and p are positive constants and p 2

There are exactly four points on C where the tangent is perpendicular to the initial line.

Show that the range of possible values for p is

Sketch the curve with equation

John digs a hole in his garden in order to make a pond.

The pond has a uniform horizontal cross section that is modelled by the curve with equation

where r is measured in centimetres.

The depth of the pond is 90 centimetres.

Water flows through a hosepipe into the pond at a rate of 50 litres per minute.

Given that the pond is initially empty,

determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute.

State a limitation of the model.

The curve C has equation

where a and p are positive constants and p 2

There are exactly four points on C where the tangent is perpendicular to the initial line.

Show that the range of possible values for p is

Sketch the curve with equation

John digs a hole in his garden in order to make a pond.

The pond has a uniform horizontal cross section that is modelled by the curve with equation

where r is measured in centimetres.

The depth of the pond is 90 centimetres.

Water flows through a hosepipe into the pond at a rate of 50 litres per minute.

Given that the pond is initially empty,

determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute.

State a limitation of the model.

Figure 1 shows a sketch of two curves C₁ and C₂ with polar equations

The region R lies inside C₁ and outside C₂ and is shown shaded in Figure 1. Show that the area of R is

where p and q are integers to be determined.

Figure 1 shows a sketch of two curves C₁ and C₂ with polar equations

The region R lies inside C₁ and outside C₂ and is shown shaded in Figure 1. Show that the area of R is

where p and q are integers to be determined.

Figure 1
Figure 2

Figure 1 shows a sketch of a design for a road speed bump of width 2.35 metres. The speed bump has a uniform cross‑section with vertical ends and its length is 30 cm. A side profile of the speed bump is shown in Figure 2.

The curve C shown in Figure 2 is modelled by the polar equation

The units for r are centimetres and the initial line lies along the road surface, which is assumed to be horizontal.

Once the speed bump has been fixed to the road, the visible surfaces of the speed bump are to be painted.

Determine, in cm², the area that is to be painted, according to the model.

Figure 1 shows the design for a table top in the shape of a rectangle ABCD. The length of the table, AB, is 1.2 m. The area inside the closed curve is made of glass and the surrounding area, shown shaded in Figure 1, is made of wood.

The perimeter of the glass is modelled by the curve with polar equation

where a is a constant.

=Show that a

Hence, given that AD = 60 cm,

find the area of the wooden part of the table top, giving your answer in m² to 3 significant figures.

Figure 1 shows the design for a table top in the shape of a rectangle ABCD. The length of the table, AB, is 1.2 m. The area inside the closed curve is made of glass and the surrounding area, shown shaded in Figure 1, is made of wood.

The perimeter of the glass is modelled by the curve with polar equation

where a is a constant.

Show that a

Hence, given that AD = 60 cm,

find the area of the wooden part of the table top, giving your answer in m² to 3 significant figures.