EDEXCEL GCSE MATHS · HIGHER
Exam Practice: Similarity in 3D Solids
Similarity in 3D solids · Similarity and congruence · Lesson 5 of 5 · 19 marks · 30 minutes
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Instructions
• Answer all the questions.
• Write your answers in the spaces provided.
• The marks for each question are shown in brackets - use this as a guide to how much to write.
• The answers are on separate pages at the back. Attempt every question before you look at them.
• Answer all questions. Show your working.
Question 1 WORK OUT (3 marks)
Cone A and cone B are mathematically similar. The height of cone A is 6 cm and its volume is 40 cm³. The height of cone B is 9 cm. Work out the volume of cone B.
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(Total for Question 1 = 3 marks)
Question 2 WORK OUT (3 marks)
A model of a statue is 20 cm tall and has a mass of 2 kg. The real statue is 1.2 m tall and is made from the same material. Work out the mass of the real statue.
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(Total for Question 2 = 3 marks)
Question 3 WORK OUT (4 marks)
A cone has a height of 15 cm and a volume of 500 cm³. A smaller cone of height 6 cm is cut off the top, leaving a frustum. The small cone is similar to the large cone. Work out the volume of the frustum.
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(Total for Question 3 = 4 marks)
Question 4 WORK OUT (3 marks)
Two similar solids have surface areas in the ratio 4 : 9. Work out the ratio of their volumes.
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(Total for Question 4 = 3 marks)
Question 5 WORK OUT (3 marks)
A small bottle is 10 cm tall and holds 250 ml of juice. A similar large bottle is 15 cm tall. Work out the capacity of the large bottle. Give your answer to the nearest ml.
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(Total for Question 5 = 3 marks)
Question 6 WORK OUT (3 marks)
Two similar solids have volumes of 8 cm³ and 125 cm³. The surface area of the smaller solid is 20 cm². Work out the surface area of the larger solid.
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(Total for Question 6 = 3 marks)
TOTAL FOR PAPER = 19 MARKS
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Answers and mark scheme
Check your answer only once you have written one.
Question 1 (3 marks)
The length scale factor is 9/6 = 1.5. The volume scale factor is 1.5³ = 3.375. The volume is 40 × 3.375 = 135 cm³.
• Length scale factor 1.5 M1
• 1.5³ = 3.375 M1
• 135 A1
Question 2 (3 marks)
The length scale factor is 120 ÷ 20 = 6. The volume scale factor is 6³ = 216. The mass is 2 × 216 = 432 kg.
• Scale factor 6 M1
• 6³ = 216 M1
• 432 A1
Question 3 (4 marks)
The length scale factor is 6/15 = 0.4 and the volume scale factor is 0.4³ = 0.064. The small cone has volume 500 × 0.064 = 32 cm³. The frustum is 500 − 32 = 468 cm³.
• Scale factor 0.4 M1
• 0.4³ = 0.064 M1
• Small cone 32 A1
• 468 A1
Question 4 (3 marks)
The length ratio is √4 : √9 = 2 : 3. The volume ratio is 2³ : 3³ = 8 : 27.
• Length ratio 2 : 3 M1
• Cubing M1
• 8 : 27 A1
Question 5 (3 marks)
The scale factor is 1.5, so the volume scale factor is 1.5³ = 3.375. The capacity is 250 × 3.375 = 843.75 ≈ 844 ml.
• Scale factor 1.5 M1
• 250 × 1.5³ M1
• 844 A1
Question 6 (3 marks)
The length ratio is ∛8 : ∛125 = 2 : 5, so the area scale factor is (5/2)² = 6.25. The surface area is 20 × 6.25 = 125 cm².
• Length scale factor 5/2 M1
• Area scale factor 6.25 M1
• 125 A1