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Measures and solids · Ages 14–15

Solids at work worksheet

Part of: Cones & spheres

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Solids at work worksheet

Measures and solids · Ages 14–15

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Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

Part A: Working out and filling in (2 marks each)

  1. 1.The cone-shaped cup in the picture is full of water; it is 8 cm in radius and 24 cm high. All the water is poured into a cylindrical cup, also 8 cm in radius. How deep is the water, in cm?

    Answer:

  2. 2.A metal ball of radius 6 cm is melted down and cast into small balls of radius 3 cm, none wasted. How many small balls does it make?

    Answer:

  3. 3.A metal ball of radius 12 cm is melted down and cast into small balls of radius 3 cm, none wasted. How many small balls does it make?

    Answer:

  4. 4.An ice cream is a cone with a hemisphere on top, both of radius 9 cm, the cone 9 cm high. What is its total volume, in terms of π?

    Answer:

  5. 5.The frustum in the picture is a cone of base radius 8 cm with its top cut off half way up: the frustum is 9 cm high and its top radius 4 cm. What is its volume, in terms of π?

    Answer:

  6. 6.The frustum in the picture has a base radius of 14 cm, a top radius of 8 cm and a height of 15 cm. Put back the cone it was cut from: the small cone cut off is cm high, and the frustum's volume, in terms of π, is π cm³.

    Fill in each blank

    Answer:

  7. 7.A sharpened pencil is a cylinder of radius 5 mm and length 13 mm with a cone-shaped tip 3 mm high on one end, of the same radius. Correct to 1 decimal place, what is the pencil's volume, in mm³?

    Answer:

  8. 8.The frustum in the picture has a base radius of 5 cm, a top radius of 2 cm and a height of 9 cm. Put back the cone it was cut from: the small cone cut off is cm high, and the frustum's volume, in terms of π, is π cm³.

    Fill in each blank

    Answer:

Part B: Choose, or true or false (2 marks each)

  1. 9.A cone of base radius 7 cm and height 16 cm is cut by a plane parallel to its base, the small cone on top 14 as high as the whole. What fraction of the whole cone's volume is the small cone?

    • A. 164
    • B. 116
    • C. 14
    • D. 364

    Choose the right answer

    Answer:

  2. 10.A cone of base radius 9 cm and height 8 cm is cut by a plane parallel to its base, the small cone on top 12 as high as the whole. What fraction of the whole cone's volume is the small cone?

    • A. 18
    • B. 14
    • C. 38
    • D. 12

    Choose the right answer

    Answer:

  3. 11.A metal ball of radius 8 cm can be melted and cast into 16 small balls of radius 2 cm, none wasted.

    True or false?

    Answer:

  4. 12.A metal ball of radius 5 cm can be melted and cast into 25 small balls of radius 1 cm, none wasted.

    True or false?

    Answer:

Part C: Word problems (4 marks each)

  1. 13.A metal cylinder of radius 12 cm and height 17 cm is melted and cast into balls of radius 1 cm, none wasted. How many balls does it make?

    Working:

    Answer:

  2. 14.A cone-shaped container full of water (5 cm in radius at the top, 60 cm deep) is emptied into a cylindrical container of base radius 10 cm. How deep is the water, in cm?

    Working:

    Answer:

  3. 15.An ice cream is a cone 5 cm in radius and 10 cm high, filled, with a hemisphere of ice cream of the same radius on top. Correct to 1 decimal place, how many cm³ of ice cream are there?

    Working:

    Answer:

  4. 16.A bucket is a frustum: 8 cm in radius at one end, 4 cm at the other, 27 cm deep (shown turned over in the picture). Made back into a cone: the whole cone is π cm³, the small cone cut off π cm³, and the bucket holds π cm³.

    Fill in each blank

    Working:

    Answer:

  5. 17.A metal cylinder of radius 12 cm and height 6 cm is melted and cast into balls of radius 3 cm, none wasted. How many balls does it make?

    Working:

    Answer:

  6. 18.A bucket is a frustum: 10 cm in radius at one end, 8 cm at the other, 12 cm deep (shown turned over in the picture). Made back into a cone: the whole cone is π cm³, the small cone cut off π cm³, and the bucket holds π cm³.

    Fill in each blank

    Working:

    Answer:

Show the answers and working

Solids at work worksheet: Answers and working

Part A: Working out and filling in

  1. 1. 8 cm

    The water is 13 × π × 82 × 24; the cylinder's base is π × 82, so the depth = volume ÷ base = 24 ÷ 3 = 8 cm.

  2. 2. 8 balls

    The volume is kept: 43 π × 63 ÷ (43 π × 33); the π and 43 cancel: 63 ÷ 33 = 8.

  3. 3. 64 balls

    The volume is kept: 43 π × 123 ÷ (43 π × 33); the π and 43 cancel: 123 ÷ 33 = 64.

  4. 4. 729 π cm³

    The cone 13 × π × 92 × 9 = 243π, the hemisphere 23 × π × 93 = 486π, 729π cm³ in all.

  5. 5. 336 π cm³

    The whole cone is 18 high: 13 × π × 82 × 18 = 384π; the small one cut off is 13 × π × 42 × 9 = 48π; the difference is 336π cm³ (78 of the whole).

  6. 6. 20, 1860

    Similar: small height : big height = 8 : 14, and the big height = the small + 15, so the small cone is 20 high and the big 35; volume 13 × π × (142 × 35 − 82 × 20) = 1860π.

  7. 7. 1099.6 mm³

    The cylinder π × 52 × 13 plus the cone 13 × π × 52 × 3, with the calculator's π: about 1099.6 mm³.

  8. 8. 6, 117

    Similar: small height : big height = 2 : 5, and the big height = the small + 9, so the small cone is 6 high and the big 15; volume 13 × π × (52 × 15 − 22 × 6) = 117π.

Part B: Choose, or true or false

  1. 9. A. 164

    The small cone is similar to the whole, lengths 1 : 4, so volumes 1 : 43, that is 164.

  2. 10. A. 18

    The small cone is similar to the whole, lengths 1 : 2, so volumes 1 : 23, that is 18.

  3. 11. False

    Volumes go as the radii cubed: (8 ÷ 2)3 = 43 = 64.

  4. 12. False

    Volumes go as the radii cubed: (5 ÷ 1)3 = 53 = 125.

Part C: Word problems

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

  1. 13. 1836 balls

    π × 122 × 17 ÷ (43 × π × 13) = 3 × 122 × 17 ÷ (4 × 13) = 1836.

  2. 14. 5 cm

    The water is 13 × π × 52 × 60; ÷ the cylinder's base π × 102 = 5 cm.

  3. 15. 523.6 cm³

    The cone 13 × π × 52 × 10 plus the hemisphere 23 × π × 53, with the calculator's π: about 523.6 cm³.

  4. 16. 1152, 144, 1008

    Similar: the small cone is 27 high and the big 54 (4 : 8); 13 × π × 82 × 54 = 1152π, 13 × π × 42 × 27 = 144π, the difference 1008π.

  5. 17. 24 balls

    π × 122 × 6 ÷ (43 × π × 33) = 3 × 122 × 6 ÷ (4 × 33) = 24.

  6. 18. 2000, 1024, 976

    Similar: the small cone is 48 high and the big 60 (8 : 10); 13 × π × 102 × 60 = 2000π, 13 × π × 82 × 48 = 1024π, the difference 976π.

Numsprite · numsprite.com/en/worksheets/cones-and-spheres/melting-pouring-and-frustums

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