Square and cubic units, and density

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   A crate holds at most 0.8 m³ and 1,200 kg. Ari fills all 0.8 m³ with sand (1,600 kg per m³) and wood chips (250 kg per m³). What is the most sand?

   • About 0.74 m³: 1,600s + 250(0.8 − s) = 1,200
   • Any amount: the average, 925 kg per m³ × 0.8, gives 740 kg
   • About 0.65 m³: 1,600s + 250s = 1,200
   • 0.75 m³: 1,200 ÷ 1,600, where sand alone reaches the limit
2. 2.

   A crown of gold (19.3 g/cm³) and silver (10.5 g/cm³) has mass 1,000 g and pushes aside 60 cm³ of water. How many cm³ of gold does Bo find?

   • 30 cm³: half, since 16.7 g/cm³ lies between the two
   • About 51.8 cm³: 1,000 ÷ 19.3, the volume of pure gold
   • About 18 cm³: 19.3(60 − g) + 10.5g = 1,000
   • About 42 cm³: 19.3g + 10.5(60 − g) = 1,000
3. 3.

   Esme ties foam (0.05 g/cm³) to a 2,000 g steel anchor (7.85 g/cm³) so the pair just floats in water (1 g/cm³). What is the least foam volume?

   • 2,000 cm³: the foam must push aside the anchor's 2,000 g
   • About 2,105 cm³: each cm³ of foam lifts 0.95 g, so 2,000 ÷ 0.95
   • About 1,745 cm³: 2,000 − 254.8, the water still to push aside
   • About 1,837 cm³: 2,000 + 0.05V = V + 254.8
4. 4.

   Jun's raft is 6 logs, each a cylinder 3 m long and 0.3 m across. Wood is 600 kg per m³; water is 1,000. What load can it carry before sinking (π = 3.14)?

   • About 763 kg: 600 kg per m³ × 1.27 m³ of logs
   • About 1,272 kg: the logs push aside 1.27 m³ of water
   • About 2,035 kg: 5.09 m³ of logs, using 0.3 m as the radius
   • About 509 kg: 1,272 kg of lift minus the logs' 763 kg
5. 5.

   A town is a disc of radius 4 km with 50,240 people. It annexes the ring out to 5 km, home to 14,130 people. What does Kira find happens to the density (π = 3.14)?

   • It falls to 750, the average of 1,000 and 500
   • It falls to 820: the ring's 500 per km² is sparser than the core
   • It rises: the town gains 14,130 people
   • It rises to about 1,206: the ring is 3.14 × (5 − 4)² km²
6. 6.

   A city's center, a disc 4 km across, has 2,000 people per km². The ring out to 5 km has 800 per km². What population does Luz estimate (π = 3.14)?

   • About 47,700: the ring is 3.14 × (5 − 2)² km²
   • About 123,100: 3.14 × 4² × 2,000 + 3.14 × (5² − 4²) × 800
   • About 77,900: 3.14 × 2² × 2,000 + 3.14 × (5² − 2²) × 800
   • About 109,900: 3.14 × 5² × 1,400, the average density
7. 7.

   A round plaza has radius 20 m. Milo paints a circle with the same center so the inner disc and the outer ring have equal areas. What radius is right?

   • About 14.1 m: 20 ÷ √2, since area scales as radius squared
   • About 15.9 m: 20 × ∛½, since the area halves
   • 10 m: half the radius gives half the area
   • 5 m: 20 × (½)², since area goes with the square
8. 8.

   As many people live within 3 km of a city's center as in the ring from 3 km to 6 km. Noah claims the ring is half as dense. Which verdict is right?

   • Wrong: π(6³ − 3³) is 7 times π × 3³, so the ring is 1/7 as dense
   • Wrong: the ring is π(6² − 3²) = 27π, 3 times the disc, so ⅓ as dense
   • Right: the outer radius doubles, so the area doubles
   • Wrong: the area grows by 2² = 4, so the ring is ¼ as dense
9. 9.

   A can (radius 4 cm, height 10 cm) uses 112π cm² of metal. Ren must double its volume. Which plan uses the least metal?

   • Height 20 cm, radius 4 cm: the lids stay the same, 192π cm²
   • Radius × √2, height 10 cm: 64π + 113π ≈ 177π cm²
   • Any plan: doubling the volume doubles the metal to 224π cm²
   • Radius 8 cm, height 10 cm: 128π + 160π = 288π cm²
10. 10.

    A 1:10 model bridge holds 3 times its own weight. Strength follows the beams' cross-section area. Zee builds the full bridge in the same material. What can it hold?

    • 0.03 times its own weight: strength × 10, weight × 1,000
    • 0.3 times its own weight: strength × 100, weight × 1,000
    • 3 times its own weight: strength and weight both grow by 100
    • 30 times its own weight: strength × 1,000, weight × 100