MAT8 Q2 W4D3: Solving Problems Involving Volumes of Cones and Spheres

🎯 Learning Goals

1. Interpret real-life and mathematical problems involving cones and spheres, identifying the given, unknown, and required steps with 90% accuracy.
2. Solve at least 4 out of 5 word problems involving the volumes of cones and spheres using the appropriate formulas within 20 minutes.
3. Explain in their own words how problem-solving strategies can be applied to different contexts involving three-dimensional objects.

🧩 Key Ideas & Terms

• Volume - the measure of space occupied by a 3D solid.
• Cone - a solid with a circular base and one vertex; volume: $\frac{1}{3}\pi r^{2}h$.
• Sphere - a perfectly round 3D shape where all points are equidistant from the center; volume: $\frac{4}{3}\pi r^3$.
• Problem-Solving Steps - Identify the given, unknown, formula, and solve with proper substitution.
• Application - the use of formulas to solve real-life and mathematical word problems.

🔄 Prior Knowledge

1. Recall the formula for the volume of a cone:
   $$V=\frac{1}{3}\pi r^{2}h$$
2. Recall the formula for the volume of a sphere:
   $$V=\frac{4}{3}\pi r^3$$
3. Quick item: A cone has a radius of 5 cm and a height of 12 cm. Find its volume.

Show Answer

$V=\frac{1}{3}\pi \left(5\right)2^{\mathrm{}}\left(12\right)$ = $\frac{300}{3}\pi$ = 100π ≈ 314.16 cm³

📖 Explore the Lesson

Why Do We Solve Word Problems?

Mathematics is not just about formulas. It is about applying concepts to real situations. Knowing the volume of cones and spheres allows us to estimate storage capacity, design sports equipment, compute food portions, solve engineering problems, and explore scientific questions.

Guiding Question: Why is it not enough to just memorize formulas for cones and spheres?

Show Answer

Because real-life problems are not given as direct formulas. We must identify what is asked, extract data, choose the correct formula, and solve step by step.

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The Problem-Solving Framework

1. Understand the Problem - read carefully, visualize, draw if needed.
2. Plan the Solution - identify given values, unknowns, and the correct formula.
3. Carry Out the Plan - substitute into the formula, compute step by step.
4. Check and Reflect - verify units, reasonableness of answer, and real-world meaning.

Checkpoint: What step ensures your answer makes sense in context?

Show Answer

The “Check and Reflect” step ensures the computed value matches the real situation logically and with correct units.

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Cones - Worked Examples

Example 1: An ice cream cone has r = 4 cm and h = 9 cm. Find its volume.

Show Solution

V = (1/3)π(16)(9) = 48π ≈ 150.8 cm³.

Example 2: A traffic cone has r = 7 cm and h = 24 cm. Find its volume.

Show Solution

V = (1/3)π(49)(24) = 392π ≈ 1232.6 cm³.

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Spheres - Worked Examples

Example 1: A basketball has r = 12 cm. Find its volume.

Show Solution

V = (4/3)π(1728) = 2304π ≈ 7238.2 cm³.

Example 2: An orange has d = 8 cm. Find its volume.

Show Solution

r = 4 cm. V = (4/3)π(64) = 256π/3 ≈ 268.1 cm³.

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Combined Cone and Sphere Problems

Example: A cone r = 4 cm, h = 9 cm holds a spherical scoop r = 4 cm on top. Find total volume.

Show Solution

Cone: 48π ≈ 150.8 cm³. Sphere: (4/3)π(64) ≈ 268.1 cm³. Total ≈ 418.9 cm³.

Checkpoint: Which contributes more volume, the scoop or the cone?

Show Answer

The scoop (sphere) contributes more volume than the cone.

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Common Errors

• Forgetting to halve diameter to get radius.
• Using the wrong formula.
• Units mismatch (cm³ vs m³, L vs mL).
• Not adding/subtracting volumes correctly in combined problems.

Checkpoint: If an answer for a small fruit is 5000 cm³, what likely went wrong?

Show Answer

Probably used diameter instead of radius and then cubed it.

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Guided Practice Activities

Activity A – Cone Problems

1. r = 6 cm, h = 12 cm.
2. r = 5 cm, h = 20 cm.

Activity B – Sphere Problems

1. r = 3.5 cm.
2. d = 50 cm.

Activity C – Combined Problems

1. Cone r = 4 cm, h = 8 cm with sphere r = 4 cm on top.
2. Hollow cone vs. sphere comparison.

Show Example Answers

Activity A: 452.4 cm³, 523.6 cm³. Activity B: 179.6 cm³, 65,449.8 cm³. Activity C: Total ≈ 402.1 cm³; ball smaller.

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Real-World Applications

1. Food industry – cone portions and spherical scoops.
2. Sports – ball volumes and air capacity.
3. Architecture – domes approximated as parts of spheres.
4. Science – modeling molecules as spheres.
5. Space – planetary volume calculations.

Checkpoint: Why do scientists need planet volumes?

Show Answer

To estimate mass, density, and gravitational effects.

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References

• Burger, E., Chard, D. J., Hall, E. J., Kennedy, P. A., & Leinwand, S. J. (2008). Holt California Geometry. Holt, Rinehart and Winston.
• Serra, M. (2008). Discovering Geometry: An Investigative Approach. Key Curriculum Press.

💡 Example in Action

5 Worked Examples

1. Cone: r = 6 cm, h = 12 cm → V ≈ 452.4 cm³.
2. Sphere: d = 20 cm → V ≈ 4188.8 cm³.
3. Conical funnel: r = 5 cm, h = 15 cm → V ≈ 392.7 cm³.
4. Orange: r = 4.5 cm → V ≈ 381.7 cm³.
5. Ice cream: cone r = 3 cm, h = 8 cm and scoop r = 3 cm → total ≈ 188.5 cm³.

Show Solutions

Shown in Day 3 examples above step by step.

📝 Now You Try

5 Practice Items

1. Cone: r = 7 cm, h = 14 cm.
2. Sphere: d = 24 cm.
3. Cone candle: r = 4 cm, h = 12 cm.
4. Bead: r = 1.2 cm.
5. Party hat + knob: cone r = 5 cm, h = 20 cm and sphere r = 5 cm.

Show Answer

1) 718.4 cm³; 2) 7238.2 cm³; 3) 201.1 cm³; 4) 7.2 cm³; 5) ≈ 1047.2 cm³.

✅ Check Yourself

10 Mixed Items (MCQ, T/F, Short Answer)

1. Which formula gives cone volume? a) πr²h b) (1/3)πr²h c) (4/3)πr³ d) (2/3)πr²h
2. T/F: A sphere’s diameter is half its radius.
3. Write the sphere volume formula.
4. A cone and cylinder with same r and h: cone volume is… a) same b) 1/2 c) 1/3 d) 2×
5. Sphere r = 9 cm. Find V.
6. T/F: Cone volume uses slant height.
7. Cone V = 314 cm³, r = 5 cm. Find h. a) 6 b) 12 c) 9 d) 15
8. Cone r = 10 cm, h = 30 cm. Find V.
9. Which is NOT a sphere? a) Basketball b) Orange c) Dice d) Marble
10. Ball r = 6 cm on cone r = 6 cm, h = 12 cm. Find total V.

Show Answer Key

1) b 2) False 3) (4/3)πr³ 4) c 5) ≈ 3053.6 cm³ 6) False 7) b 8) 1000π ≈ 3141.6 cm³ 9) c 10) ≈ 1357.2 cm³

🚀 Go Further

Day 3 - 5 Activities

Activity 1 – Real-Life Cone vs. Sphere – find one example of each at home, measure/estimate, compute, and compare volumes.

Show Example

Cone (funnel: r = 5 cm, h = 15 cm → V ≈ 392.7 cm³). Sphere (ball: r = 6 cm → V ≈ 904.8 cm³). Sphere is about twice the cone volume.

Activity 2 – Sports Application – compare volumes of tennis ball (r = 3.5 cm) and basketball (r = 12 cm); estimate how many tennis balls equal one basketball volume.

Show Answer

Tennis ball ≈ 179.6 cm³, basketball ≈ 7238.2 cm³. About 40 tennis balls equal one basketball volume.

Activity 3 – Engineering Task – conical tank (r = 2 m, h = 4 m) filled with spherical balls (r = 0.5 m). How many balls? (ignore packing gaps)

Show Answer

Cone ≈ 16.76 m³; sphere ≈ 0.52 m³; maximum ≈ 32 balls.

Activity 4 – Creative Challenge – design a cone + sphere figure with labeled dimensions and compute total volume.

Show Example

Cone r = 4 cm, h = 10 cm → 167.6 cm³; sphere r = 4 cm → 268.1 cm³; total 435.7 cm³.

Activity 5 – Research & Connect – explain why spherical tanks store gases well and why conical silos are used for grains.

Show Example

Spheres distribute pressure evenly; cones aid discharge by gravity.

🔗 My Reflection

Option A – Write 3–5 Sentences

In my notebook, I will write a reflection on today’s lesson. I will explain how I solved problems involving cones and spheres, what strategies helped me avoid mistakes, and how these connect to real-life contexts such as sports, food, or engineering.

Option B – Checklist

• ☐ I can identify the correct formula for cones and spheres.
• ☐ I can solve real-life problems involving volumes of cones and spheres.
• ☐ I can compare volumes of cones and spheres accurately.
• ☐ I can apply the problem-solving steps (Understand, Plan, Solve, Check).
• ☐ I can explain why cones and spheres are used in specific industries.

Instruction: Answer these in your notebook.