Square and cubic units, and density

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

   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
   • 10 × 10 × 10 cm: a cube has the least surface, so its $10.00 is lowest
   • 5 × 5 × 25.6 cm: the cheapest of the four, just $6.62
   • 6.25 × 6.25 × 25.6 cm: $8.74, as costly lids favor a smaller base
2. 2.

   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 high by 18π ≈ 57 m³: a dome fills ⅓ of its cylinder, like a cone
   • Too high by 27π ≈ 85 m³: the top 3 m of the model is all empty
   • 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

3. Read this first

   Step 1

   Ring area = π(0.5 − 0.4)² ≈ 0.0314 m². Step 2: volume = 0.0314 × 2 = 0.0628 m³. Step 3: mass = 0.0628 × 2,400 ≈ 151 kg, so the 500 kg crane can lift it.

4. 3.

   A concrete pipe is 2 m long, with outer radius 0.5 m and inner radius 0.4 m. Concrete is 2,400 kg per m³. Jun checks this model. Which correction is right?

   • Step 1: use the outer radius, π × 0.5² ≈ 0.785 m², so about 3,768 kg
   • Step 3: mass is volume ÷ density, so 0.0628 ÷ 2,400 kg
   • No step is wrong: 151 kg is under 500 kg, so the crane can lift it
   • Step 1: the ring is π(0.5² − 0.4²) ≈ 0.283 m², so about 1,356 kg
5. 4.

   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)?

   • No: dry land is 4,000 − 314 = 3,686 m², which allows 7,372
   • 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
   • Yes: the pond takes 2 × 3.14 × 10 = 62.8 m², leaving room for 7,874
6. 5.

   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?

   • 1,500: the floor is 20 × 15 = 300 ft², at 5 BTU each
   • 15,000: 2,400 ft³ plus a roof prism of ½ × 15 × 4 × 20 = 600 ft³
   • 18,000: the roof space adds 15 × 4 × 20 = 1,200 ft³
   • 12,000: the room is 20 × 15 × 8 = 2,400 ft³, at 5 BTU each
7. 6.

   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 18,000 more people than A
   • 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 440 per mi² against A's 400

8. District

   21,600 people. Step 1: area = 12 × 9 = 108 km². Step 2: density = 21,600 ÷ 108 = 200 per km². Step 3: 200 < 380, so no clinic is needed.

9. 7.

   A district is a triangle with corners (0, 0), (12, 0) and (4, 9) in km. Above 380 people per km², it needs a second clinic. Which fix to Noah's work is right?

   • Step 1: the height is the slanted side, √97 ≈ 9.85 km, so about 365: no clinic
   • Step 1: a triangle is ½ × 12 × 9 = 54 km², so 400 per km²: a clinic
   • No step is wrong: the corners span 12 km by 9 km, so 108 km² is right
   • Step 2: density is area per person, 108 ÷ 21,600 = 0.005
10. 8.

    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?

    • 1,350 g of batter and 506.25 g of frosting
    • 900 g of batter and 506.25 g of frosting
    • 900 g of batter and 337.5 g of frosting
    • 1,350 g of batter and 337.5 g of frosting
11. 9.

    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 ÷ 1.6² = 1,250 per km², under Q's 1,300
    • City P: 3,200 ÷ 1.6 = 2,000 per km², above Q's 1,300
    • City P: 3,200 × 1.6² ≈ 8,192 per km², far above 1,300
    • City Q: 3,200 ÷ 1.6³ ≈ 781 per km², far under 1,300
12. 10.

    Zee models a tapering log with two cylinders before a crane lifts it. Which conclusion do the two models justify?

    • Safe: 1,618 kg is just 8% over, inside a crane's margin
    • Safe: Model 2 shows the log is 856 kg, well under
    • Unsafe: Model 1 shows the log is 1,618 kg, over 1,500
    • Not settled: the mass is between 856 and 1,618 kg, and 1,500 lies between