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 parallel cut makes the cones similar, so heights match radii
   • The two cones have the same height, so their height ratio is 1 and nothing is left to find
   • The bucket looks like a cone with straight sides, so the ratio holds by eye alone
   • The bucket's average radius, 4.5, gives the small cone's height after a rescale
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 looks small on the page, about one tenth of the whole cone by eye
   • The top cone's base area is 1/9 of the whole, so its volume is also 1/9 of it
   • 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
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 come to a point at the top, so their volumes must be equal by that shape
   • Both have the same slant height, so their volumes are equal because slant decides
   • Both have base area 30, and the base area alone decides the volume of a solid
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 area of a circle 10 cm in radius times the rise
   • The rock's volume equals the base area times the new depth of the water
   • 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?

   • Yes: both have straight sides and the same height, so they hold the same
   • Yes: the halfway radius gives the halfway volume of the two cones
   • No: the can is 192π, the volume of the big cone alone, and the bucket is less
   • 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?

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

   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 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
   • No: the ball is π/4 of the cylinder, like a circle in a square, so under 80%
9. 8.

   Pia finds a pyramid's volume (base 6 cm square, slant height 5 cm): Line 1: height = 5 − 3 = 2. Line 2: volume = (1/3)(36)(2) = 24. Which line is wrong?

   • Neither line is wrong; just the units are missing from the final answer
   • Line 2: the pyramid formula has no one third, so the volume is 72
   • Line 2: the base area should be 6 × 3 = 18, using half of one side
   • Line 1: the height is √(5² − 3²) = 4 by Pythagoras, not 5 − 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.

    Zee finds a cone's height (radius 5 cm, volume 100π cm³): Line 1: (1/3)π·25·h = 100π. Line 2: 25h = 100. Line 3: h = 4. Which line is wrong?

    • Line 3: divide 25h = 100 by 25π, because the π was not removed
    • Line 1: the cone formula has no one third, so the equation is wrong from the start
    • Line 1: the radius should be 10 cm, as the cone base is wide
    • Line 2: times 3 gives 25h = 300, so h = 300 ÷ 25 = 12