Volume of combined solids

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   A can of radius 4 cm and height 10 cm sits in a box 8 cm by 8 cm by 10 cm. Bo finds the share of the box the can fills. What is it?

   • About 78.5%: 160π ÷ 640 = π/4
   • About 26%: the can taken as a cone
   • About 50%: half the box
   • 100%: the can touches the four walls
2. 2.

   A ball of radius r fits snugly inside a cylinder of radius r and height 2r. Esme finds the share of the cylinder the ball fills. What is it?

   • 1/2: the ball looks like half
   • π/4: the ratio of a circle to its square
   • 1/3: the ball taken as a cone
   • 2/3: (4/3)πr³ ÷ 2πr³
3. 3.

   A cone and a half-ball both have radius r, and the cone is r tall. Jun compares their volumes. What is the ratio of cone to half-ball?

   • 1 to 1: both fit in the same cylinder
   • 1 to 2: (1/3)πr³ and (2/3)πr³
   • 1 to 4: a full ball used
   • 3 to 2: the cone taken as πr³
4. 4.

   Two cones are similar. Their heights are 6 cm and 9 cm, and the small one holds 40 mL. Kira finds the big one's capacity. What is it?

   • 11.9 mL: 40 × (6/9)³
   • 60 mL: 40 × 9/6
   • 90 mL: 40 × (9/6)²
   • 135 mL: 40 × (9/6)³
5. 5.

   Sketch, not to scale. A 1:25 scale model of a pool holds 64 liters. Luz finds the real pool's volume in liters. What is it?

   • 1,000,000 liters: 64 × 25³
   • 1,600 liters: 64 × 25, the length factor
   • 40,000 liters: 64 × 25²
   • 0.004 liters: 64 ÷ 25³
6. 6.

   Two bottles are similar; the big one is 1.5 times as tall. Milo compares their label areas and capacities. By what factors do they grow?

   • Label area × 2.25, capacity × 2.25
   • Label area × 1.5, capacity × 1.5
   • Label area × 2.25, capacity × 3.375
   • Label area × 3.375, capacity × 2.25
7. 7.

   Sketch, not to scale. A cone-shaped cup is filled with water to half its height. Noah finds the share of the cup's volume that is water. What is it?

   • 1/4: half the radius squared
   • 7/8: the empty top compared
   • 1/8: a similar cone, half as tall
   • 1/2: half the height, so half the volume
8. 8.

   A right triangle with legs 3 and 4 spins around one of its legs. Pia picks the spin that makes the bigger cone. Which is it?

   • Spin about the hypotenuse 5: the longest side gives the biggest cone
   • Equal: it is the same triangle both times
   • Spin about the 3 leg: (1/3)π·4²·3 = 16π against 12π
   • Spin about the 4 leg: radius 4 and height 3 give 16π against 12π

9. Read this first

   Use this

   Sketch, not to scale.

10. 9.

    A sphere has radius 13 cm. A flat cut 5 cm from its center makes a circle. Tala finds the circle's area. What is it?

    • 25π cm²: radius 5
    • 194π cm²: radius √(13² + 5²)
    • 144π cm²: radius √(13² − 5²) = 12
    • 169π cm²: the sphere's biggest circle
11. 10.

    A can has radius 6 cm and height 6 cm. Zee drills out a cone with the same radius and height. What is left, compared with a half-ball of radius 6 cm?

    • 144π cm³, equal to the half-ball's (2/3)π·6³
    • 288π cm³: the half-ball taken as a full ball
    • 0 cm³: the cone taken as π·6²·6
    • 216π cm³: the cone left in