Volume of combined solids

Grade 10 Math Worksheet

Circle the best answer.

1. Read this first

   Use this

   Sketch, not to scale.

2. 1.

   Ari finds a bucket's volume (radii 6 and 3 cm, height 8 cm) as a big cone minus a small cone, using 1/2 = h/(h + 8). What licenses that ratio?

   • The bucket looks like a cone with straight sides, so the ratio holds by eye alone
   • The two cones have the same height, so their height ratio is 1 and nothing is left to find
   • The bucket's average radius, 4.5, gives the small cone's height after a rescale
   • The parallel cut makes the cones similar, so heights match radii
3. 2.

   A cone is 12 cm tall with radius 6 cm. Bo cuts off the top 4 cm parallel to the base and says the top cone is 1/27 of the whole. What licenses 1/27?

   • The top cone is similar with scale 1/3, so its volume is 1/3 of the whole cone
   • It is similar with scale 1/3, so volume scales by (1/3)³
   • The top cone's base area is 1/9 of the whole, so its volume is also 1/9 of it
   • The top looks small on the page, about one tenth of the whole cone by eye
4. 3.

   A cone and a square pyramid both have base area 30 cm² and height 12 cm. Esme wants to show equal volumes without using volume formulas. Which argument works?

   • Equal slice areas at every height, so Cavalieri gives equal volumes
   • Both have base area 30, and the base area alone decides the volume of a solid
   • Both have the same slant height, so their volumes are equal because slant decides
   • Both come to a point at the top, so their volumes must be equal by that shape
5. 4.

   A rock is lowered into a cylinder 10 cm across. The water level rises 2 cm. Sketch, not to scale. A rock sinks into a cylinder 10 cm across and the water rises 2 cm. Jun says the rock's volume is the base area times the rise. What licenses that?

   • The rock's volume equals one third of the base area times the rise, as for a cone
   • The rock's volume equals the base area times the new depth of the water
   • The rock's volume equals the area of a circle 10 cm in radius times the rise
   • It pushes aside equal water: the tank's base times the rise
6. 5.

   Kira says a bucket with radii 6 and 3 cm and height 8 cm has the same volume as a can of average radius 4.5 cm and height 8 cm. Which reply is right?

   • No: the can is 192π, the volume of the big cone alone, and the bucket is less
   • Yes: the halfway radius gives the halfway volume of the two cones
   • Yes: both have straight sides and the same height, so they hold the same
   • No: the bucket is 168π and the can is 162π, since volume follows r²
7. 6.

   Luz says doubling every length of a cone doubles its volume. Which reply is right?

   • Yes: each length doubled, so the volume doubles
   • No: it becomes 6 times as large, from 3 × 2
   • No: it becomes 4 times as large, like the base area
   • No: volume scales with the cube, so 8 times as large
8. 7.

   Milo says two solids with equal height and equal base area always have equal volume. Which reply is right?

   • Yes: the same base and the same height make the same volume for every solid
   • No: they match only if their slant heights match as well as the base
   • No: Cavalieri needs equal slices at every height; cylinder and cone differ 3 to 1
   • Yes: each is one third of base times height, so they must agree anyway
9. 8.

   Noah says a ball of radius 3 cm is exactly two thirds of the snug cylinder around it (radius 3, height 6). Which reply is right?

   • No: the ball is π/4 of the cylinder, like a circle in a square, so under 80%
   • No: the ball is one third of the cylinder, like a cone in its can, so less
   • Yes: the ball is half the cylinder, which looks right
   • Yes: the ball is 36π, the cylinder is 54π, and 36π/54π = 2/3
10. 9.

    Ren finds a silo's volume (cylinder radius 3 m, height 10 m, half-ball top): Line 1: cylinder 90π. Line 2: half-ball (4/3)π·27 = 36π. Line 3: total 126π. Which line is wrong?

    • No line is wrong; just the units are missing from the total
    • Line 3: add another 36π for a second half-ball at the bottom of the silo
    • Line 2: a half-ball is half a ball, (2/3)π·27 = 18π
    • Line 1: the cylinder should use radius 6 because the silo is 6 m across
11. 10.

    Tala finds a rock's volume (cylinder 10 cm across, water from 10 cm to 12 cm): Line 1: rise = 2 cm. Line 2: volume = π·5²·12. Which line is wrong?

    • Line 2: the volume uses the rise, 2 cm, not the depth, 12 cm
    • Line 2: the radius should be 10 cm, because the cylinder is 10 cm across
    • Line 2: use π·5·2, with the radius not squared, since the rock is solid
    • Line 1: the rise should be 12 cm, the depth of the water after the rock