A LevelFurther MathsEdexcelCore PureExam QuestionsFurther CalculusVolumes of Revolution

Volumes of Revolution (Edexcel A Level Further Maths: Core Pure): Exam Questions

Exam code: 9FM0

30 mins3 questions
1a1 mark
qp8a-9fm0_02_2019

Figure 1 shows the central vertical cross section A B C D of a paddling pool that has a circular horizontal cross section. Measurements of the diameters of the top and bottom of the paddling pool have been taken in order to estimate the volume of water that the paddling pool can contain.

Using these measurements, the curve B D is modelled by the equation

y space equals space ln space left parenthesis 3.6 x space – space k right parenthesis space space space space space space space space space space space space space 1 space less or equal than space x space less or equal than space 1.18

as shown in Figure 2.

Find the value of k.

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1b
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Find the depth of the paddling pool according to this model.

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1c
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The pool is being filled with water from a tap.

Find, in terms of h, the volume of water in the pool when the pool is filled to a depth of h space straight m.

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1d
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Given that the pool is being filled at a constant rate of 15 litres every minute, find, in cm space straight h to the power of negative 1 end exponent, the rate at which the water level is rising in the pool when the depth of the water is 0.2 space straight m.

How did you do?

2a2 marks
qp7a-9fm0_02-june-2020

Figure 1

A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student’s design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the y-axis, the x-axis, the line with equation x equals 1, the curve C subscript 1 and the curve C subscript 2 through 360° about the y-axis. The point A has coordinates (1, 0.5) and the point B has coordinates (0.5, 2.5) where the units are centimetres.

The curve C subscript 1 is modelled by the equation

x equals fraction numerator a over denominator y plus b end fraction space space space space space space space space 0.5 less or equal than y less or equal than 2.5

Determine the value of a and the value of b according to the model.

How did you do?

2b
Sme Calculator9 marks

The curve C subscript 2 is modelled to be an arc of the circle with centre (0, 3).

Use calculus to determine the volume of plastic required to make the chess piece according to the model.

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3a4 marks
qp2a-9fm0-01-further-maths
Figure 2

Figure 2 shows the image of a gold pendant which has height 2cm. The pendant is modelled by a solid of revolution of a curve C about the y-axis. The curve C has parametric equations

x equals cos space theta plus 1 half sin space 2 theta comma space space space space y equals negative left parenthesis 1 plus sin theta right parenthesis space space space space space space 0 less or equal than theta less or equal than 2 pi

Show that a Cartesian equation of the curve C is

x squared equals negative left parenthesis y to the power of 4 plus 2 y cubed right parenthesis

How did you do?

3b
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Hence, using the model, find, in cm3, the volume of the pendant.

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