Square and cubic units, and density
Grade 10 Math Worksheet
Circle the best answer.
A box with a square base and a lid must hold 1,000 cm³. Base and lid cost 3¢ per cm²; the four sides cost 1¢ per cm². Which design should Ari pick?
- 5 × 5 × 40 cm: the smallest base gives the lowest cost, $9.50
- 6.25 × 6.25 × 25.6 cm: $8.74, as costly lids favor a smaller base
- 5 × 5 × 25.6 cm: the cheapest of the four, just $6.62
- 10 × 10 × 10 cm: a cube has the least surface, so its $10.00 is lowest
A crate holds at most 0.8 m³ and at most 1,200 kg. Bo fills it with either sand (1,600 kg per m³) or wood chips (250 kg per m³). Which limit stops each load?
- Sand may fill 1,600 ÷ 1,200 ≈ 1.33 m³, so space stops both loads
- Sand stops at 1,200 kg with 0.75 m³ in; chips fill the 0.8 m³ at 200 kg
- Mass stops both loads: sand at 0.75 m³ and chips at 4.8 m³
- Space stops both loads, so the full crate takes 1,280 kg of sand
A silo is a cylinder (radius 3 m, height 10 m) with a half-sphere roof of radius 3 m. Esme models the grain space as one cylinder 13 m tall. How good is the model?
- Too low: the dome adds room that a flat-topped cylinder leaves out
- Too high by 9π ≈ 28 m³: the dome fills only ⅔ of its 3 m of cylinder
- Too high by 27π ≈ 85 m³: the top 3 m of the model is all empty
- Too high by 18π ≈ 57 m³: a dome fills ⅓ of its cylinder, like a cone
A field is 80 m by 50 m with a round pond of radius 10 m. Safety allows 2 people per m² of dry land. Can Kira's fair sell 7,500 tickets (π = 3.14)?
- Yes: 80 × 50 × 2 = 8,000 people may come
- Yes: 7,500 ÷ 3,686 ≈ 2.03 per m², close enough to the limit of 2
- No: dry land is 4,000 − 314 = 3,686 m², which allows 7,372
- Yes: the pond takes 2 × 3.14 × 10 = 62.8 m², leaving room for 7,874
Luz's family heats a room with a peaked ceiling. It needs 5 BTU per hour for each cubic foot of air. Which heater on the card fits?
- 12,000: the room is 20 × 15 × 8 = 2,400 ft³, at 5 BTU each
- 1,500: the floor is 20 × 15 = 300 ft², at 5 BTU each
- 18,000: the roof space adds 15 × 4 × 20 = 1,200 ft³
- 15,000: 2,400 ft³ plus a roof prism of ½ × 15 × 4 × 20 = 600 ft³
County A has 48,000 people on 120 mi², of which 40 mi² is lake. County B has 66,000 people on 150 mi² of land. A report calls B more crowded. Which judgment should Milo make?
- The report is right: B has 440 per mi² against A's 400
- On land A is denser, 600 to 440 per mi²; the report counted A's lake
- The report is wrong: A has 1,200 per mi², from 48,000 ÷ 40
- The report is right: B has 18,000 more people than A
A round cake 20 cm across uses 400 g of batter and 150 g of frosting. Pia bakes a similar cake 30 cm across, every length scaled alike. What does it need?
- 900 g of batter and 506.25 g of frosting
- 1,350 g of batter and 506.25 g of frosting
- 1,350 g of batter and 337.5 g of frosting
- 900 g of batter and 337.5 g of frosting
A 1:4 model of a bronze statue weighs 5 kg and took 0.3 L of paint. Ren claims the full statue needs 1.2 L of paint and weighs 20 kg. Which verdict is right?
- Both are low: paint × 64 = 19.2 L and mass × 16 = 80 kg
- Both are low: paint × 16 = 4.8 L and mass × 64 = 320 kg
- Right on both: each length is 4 times, so each amount is 4 times
- Both are low: paint × 16 = 4.8 L and mass × 16 = 80 kg
City P has 3,200 people per square mile. City Q has 1,300 per square km. One mile is 1.6 km. Tala decides which city is denser. Which reasoning is right?
- City Q: 3,200 ÷ 1.6³ ≈ 781 per km², far under 1,300
- City Q: 3,200 ÷ 1.6² = 1,250 per km², under Q's 1,300
- City P: 3,200 × 1.6² ≈ 8,192 per km², far above 1,300
- City P: 3,200 ÷ 1.6 = 2,000 per km², above Q's 1,300
Zee models a tapering log with two cylinders before a crane lifts it. Which conclusion do the two models justify?
- Safe: Model 2 shows the log is 856 kg, well under
- Safe: 1,618 kg is just 8% over, inside a crane's margin
- Not settled: the mass is between 856 and 1,618 kg, and 1,500 lies between
- Unsafe: Model 1 shows the log is 1,618 kg, over 1,500