1. To determine which can of pizza sauce is the better buy, calculate the volume of each can and then find the cost per unit volume.
- Volume of a cylinder = π × r² × h
- For the first can: radius = 10 cm / 2 = 5 cm, height = 20.5 cm
- Volume = π × (5)² × 20.5 = π × 25 × 20.5 = 512.5π cm³ ≈ 1609.9 cm³
- Cost per cm³ = $3.49 / 1609.9 ≈ $0.00217
- For the second can: radius = 7.5 cm / 2 = 3.75 cm, height = 11 cm
- Volume = π × (3.75)² × 11 = π × 14.0625 × 11 = 154.6875π cm³ ≈ 486.0 cm³
- Cost per cm³ = $1.09 / 486.0 ≈ $0.00224
- The first can has a lower cost per cm³, so it is the better buy.
2. To calculate the wasted area between the cans in the box:
- The box is 18 cm by 24 cm.
- The cans are arranged in a grid. From the diagram, there are 6 cans across (18 cm / 3 cans = 6 cm diameter each) and 4 cans down (24 cm / 4 cans = 6 cm diameter each). So each can has a diameter of 6 cm.
- Area of the box = 18 cm × 24 cm = 432 cm²
- Area of one can (circle) = π × r² = π × (3 cm)² = 9π cm² ≈ 28.27 cm²
- Total area of 24 cans = 24 × 28.27 cm² ≈ 678.48 cm² (Wait, this is larger than the box area. That can't be right.)
- Let's recheck: The diagram shows 6 cans across and 4 cans down, so 6 × 4 = 24 cans. Each can has a diameter of 6 cm (since 18 cm / 3 = 6 cm? Wait, 18 cm / 6 cans = 3 cm diameter? That doesn't make sense because the cans are shown as larger. Let's look again.
- Actually, the diagram shows 6 cans across and 4 cans down. The width is 18 cm, so each can has a diameter of 18 cm / 6 = 3 cm. Height is 24 cm / 4 = 6 cm. But that would make the cans 3 cm diameter, which seems small. But let's go with the diagram.
- Diameter = 3 cm, so radius = 1.5 cm
- Area of one can = π × (1.5)² = 2.25π cm² ≈ 7.07 cm²
- Total area of 24 cans = 24 × 7.07 cm² ≈ 169.68 cm²
- Area of the box = 18 cm × 24 cm = 432 cm²
- Wasted area = 432 cm² - 169.68 cm² ≈ 262.32 cm²
3. To find how much water will be left in the cubic tank after draining into the cylindrical tank:
- Volume of the cubic tank = edge³ = (4.6 m)³ = 97.336 m³
- Volume of the cylindrical tank = π × r² × h = π × (2.2 m)² × (4.6 m) = π × 4.84 × 4.6 = 22.264π m³ ≈ 69.94 m³
- Water left in the cubic tank = 97.336 m³ - 69.94 m³ ≈ 27.396 m³
4. To find how much cat food fits into the can and how much paper is needed for the label:
- Volume of the can (cylinder) = π × r² × h
- Radius = 9 cm / 2 = 4.5 cm, height = 14.5 cm
- Volume = π × (4.5)² × 14.5 = π × 20.25 × 14.5 = 293.625π cm³ ≈ 922.3 cm³
- Surface area of the label (lateral surface area of the cylinder) = 2πrh
- = 2 × π × 4.5 cm × 14.5 cm = 130.5π cm² ≈ 410.0 cm²
Final Answers:
1. The first can is the better buy.
2. The wasted area is approximately 262.32 cm².
3. Approximately 27.396 m³ of water will be left in the tank.
4. The can can hold approximately 922.3 cm³ of cat food, and 410.0 cm² of paper is needed for the label.
Parent Tip: Review the logic above to help your child master the concept of surface area and volume word problems worksheet with answers.
