Solving Frustum and Bucket Problems

In our previous chapter, "Understanding the Frustum of a Cone", we did the heavy lifting of learning what a frustum is and where its formulas come from. We learned how to calculate its Volume, Curved Surface Area (CSA), and Total Surface Area (TSA).

But exams rarely ask you to "find the volume of a frustum" directly. Instead, they hand you a bucket, a drinking glass, a lampshade, or a Turkish cap, and ask you to find how much water it holds or how much metal sheet was used to make it.

This chapter is all about bridging the gap between the raw formulas and these classic real-world word problems. We are going to break down exactly how to approach them, how to avoid the most common traps, and how to handle the unit conversions that trip up so many students.

The Anatomy of a Real-World Frustum

When you read a word problem, your first job is to translate the everyday object into mathematical terms. Almost all frustum word problems revolve around a few standard objects. Let us look at how to interpret them.

1. The Bucket or Drinking Glass

A standard bucket or drinking glass is simply an inverted frustum of a cone.

• The top is open and has a larger radius (r₁).
• The bottom is closed (so the water doesn't fall out!) and has a smaller radius (r₂).
• The height (h) is the straight vertical depth of the bucket.
• The slant height (l) is the sloping side of the bucket.

2. The Turkish Cap (Fez)

A fez is a traditional hat shaped like a frustum.

• The bottom is open (so you can put it on your head) and has a larger radius (r₁).
• The top is closed (a flat circular piece of fabric) and has a smaller radius (r₂).

3. The Lampshade

A lampshade is a frustum designed to let light out of both ends.

• The top is completely open.
• The bottom is completely open.

The Golden Rule of Surface Area Problems

When a question asks for the "cost of the metal sheet required to make a bucket" or the "area of canvas used to make a cap," you are dealing with Surface Area.

Do not use the standard TSA formula: π l(r₁ + r₂) + π r₁² + π r₂². Instead, build your formula logically based on the object:

• For a Bucket / Drinking Glass: You need the curved sides PLUS the closed bottom.

Area = CSA + Area of smaller base Area = π l(r₁ + r₂) + π r₂²

• For a Turkish Cap (Fez): You need the curved sides PLUS the closed top.

Area = CSA + Area of smaller base Area = π l(r₁ + r₂) + π r₂²

• For a Lampshade: You only need the curved sides. Both ends are open.

Area = CSA only Area = π l(r₁ + r₂)

Mastering Unit Conversions

Before we dive into the math, we must address the final hurdle of these problems: units.

Often, a problem will give you dimensions in centimeters (cm) but ask for the capacity of the bucket in liters. Or, it will ask for the cost of the metal sheet per 100 cm². If you do not convert your final answer correctly, all your hard work on the formulas will be wasted.

Here is your essential conversion cheat sheet for mensuration problems:

• Volume (cm to Liters): Divide your answer in cm³ by 1000.

(Example: 5500 cm³ = 5.5 Liters)

• Volume (m to Liters): Multiply your answer in m³ by 1000.

(Imagine a giant water tank 1 meter on all sides. It holds 1000 liters!)

• Area (Cost calculations): If the cost is "₹20 per 100 cm²", you must divide your total area by 100 before multiplying by 20.

The Step-by-Step Problem Solving Method

When you face a frustum word problem, do not just start plugging numbers into a calculator. Follow this exact sequence to ensure you don't miss a step.

Let us put this method into practice with three classic, fully worked-out examples.

Worked Example 1: The Drinking Glass (Volume)

The Problem: A drinking glass is in the shape of a frustum of a cone of height 14 cm. The radii of its two circular ends are 6 cm and 3 cm. Find the capacity of the glass. (Take π = 22⁄7)

Step-by-Step Solution:

1. 1Identify the given values:

• Height (h) = 14 cm
• Larger radius (r₁) = 6 cm (the top of the glass)
• Smaller radius (r₂) = 3 cm (the bottom of the glass)

1. 1Identify the goal:

• "Capacity" means we need to find the Volume.

1. 1Write the formula:

• Volume of a frustum V = 1⁄3 π h (r₁² + r₂² + r₁ r₂)

1. 1Plug in the numbers and solve:

• V = 1⁄3 × 22⁄7 × 14 × (6² + 3² + 6 × 3)

Pro-tip: Always simplify the fractions outside the bracket first if possible. Here, the 7 in the denominator perfectly cancels out the 14.

• V = 1⁄3 × 22 × 2 × (36 + 9 + 18)
• V = 44⁄3 × (63)

Now, simplify the 3 and the 63.

• V = 44 × 21
• V = 924 cm³

Final Answer: The capacity of the drinking glass is 924 cm³. (If the question asked for liters, we would divide by 1000 to get 0.924 Liters).

Worked Example 2: The Metal Bucket (Surface Area & Cost)

The Problem: A container, opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm, respectively. Find the cost of the metal sheet used to make the container, if it costs ₹20 per 100 cm². (Take π = 3.14)

Step-by-Step Solution:

1. 1Identify the given values:

• Height (h) = 16 cm
• Larger radius (r₁) = 20 cm (upper end)
• Smaller radius (r₂) = 8 cm (lower end)

1. 1Identify the goal:

• We need the "cost of the metal sheet", which means we first need to find the Surface Area.
• Because it is a bucket (open at the top, closed at the bottom), the area formula is: CSA + Area of the smaller base.

1. 1Find the slant height (l):

• We cannot find CSA without l. Let's calculate it.
• l = √(h² + (r₁ - r₂)²)
• l = √(16² + (20 - 8)²)
• l = √(16² + 12²)
• l = √(256 + 144)
• l = √(400)
• l = 20 cm

1. 1Calculate the Surface Area:

• Area = π l(r₁ + r₂) + π r₂²

Pro-tip: Factor out π to make the math much cleaner!

• Area = π × [ l(r₁ + r₂) + r₂² ]
• Area = 3.14 × [ 20(20 + 8) + 8² ]
• Area = 3.14 × [ 20(28) + 64 ]
• Area = 3.14 × [ 560 + 64 ]
• Area = 3.14 × 624
• Area = 1959.36 cm²

1. 1Calculate the Cost:

• The cost is ₹20 for every 100 cm².
• This means the cost for 1 cm² is 20⁄100 = ₹0.20.
• Total Cost = Total Area × Cost per cm²
• Total Cost = 1959.36 × 0.20
• Total Cost = 391.872

Final Answer: The cost of the metal sheet required to make the bucket is ₹391.87.

Worked Example 3: The Turkish Cap (Fez)

The Problem: A fez, the cap used by the Turks, is shaped like the frustum of a cone. If its radius on the open side is 10 cm, radius at the upper base is 4 cm, and its slant height is 15 cm, find the area of material used for making it. (Take π = 22⁄7)

Step-by-Step Solution:

1. 1Identify the given values:

• Slant height (l) = 15 cm (Notice they gave us l directly! We don't need to calculate it from h.)
• Larger radius (r₁) = 10 cm (the open bottom where the head goes)
• Smaller radius (r₂) = 4 cm (the closed flat top)

1. 1Identify the goal:

• "Area of material used" means Surface Area.
• A cap is open at the bottom and closed at the top. So, we need the CSA + Area of the smaller top base.

1. 1Calculate the Surface Area:

• Area = π l(r₁ + r₂) + π r₂²
• Area = π × [ l(r₁ + r₂) + r₂² ]
• Area = 22⁄7 × [ 15(10 + 4) + 4² ]
• Area = 22⁄7 × [ 15(14) + 16 ]
• Area = 22⁄7 × [ 210 + 16 ]
• Area = 22⁄7 × 226
• Area = 4972⁄7
• Area = 710.28 cm² (or as a mixed fraction, 710 2⁄7 cm²)

Final Answer: The area of the material used for making the cap is 710.28 cm².

Common Calculation Traps & Pro-Tips

When you are under exam pressure, it is rarely the formulas that let you down—it is the arithmetic. Here is how to protect yourself from silly mistakes:

1. 1Never multiply π too early: As you saw in the examples above, always factor π out of your surface area equations: π [ l(r₁ + r₂) + r₂² ]. Solve everything inside the square brackets first. Multiplying by 3.14 or 22/7 should be the absolute last step you do.
2. 2Watch out for Diameters: Examiners love to give you the diameter of the bases instead of the radius. If a question says "the diameters of its two circular ends are 4 cm and 2 cm", immediately cross those numbers out and write r₁ = 2 and r₂ = 1 on your rough sheet.
3. 3Check the π instruction: If the question explicitly says "(Take π = 3.14)", use it. If it doesn't, use 22⁄7. Usually, the numbers in the problem are specifically chosen to cancel out perfectly with the 7 in the denominator of 22⁄7.
4. 4BODMAS in the Volume Formula: In the volume formula V = 1⁄3 π h (r₁² + r₂² + r₁ r₂), always calculate the three terms inside the bracket first, add them together, and then multiply by the outside terms. Do not try to multiply the height into the bracket prematurely.

Quick Revision

Before you walk into the exam hall, make sure these key points are locked in your memory:

• Bucket / Glass Volume (Capacity): V = 1⁄3 π h (r₁² + r₂² + r₁ r₂)
• Bucket / Glass Surface Area: Open top, closed bottom. Area = π l(r₁ + r₂) + π r₂² (where r₂ is the smaller bottom base).
• Turkish Cap (Fez) Surface Area: Open bottom, closed top. Area = π l(r₁ + r₂) + π r₂² (where r₂ is the smaller top base).
• Slant Height: If you are only given vertical height (h) and need surface area, you must find l first using l = √(h² + (r₁ - r₂)²).
• Volume to Liters: Divide cm³ by 1000 to get Liters. (1000 cm³ = 1 Liter).
• Cost Calculations: Always check the unit of cost. If it is per 100 cm², divide your total area by 100 before multiplying by the rate.
• Factoring: Always factor out π in surface area problems to save time and reduce calculation errors.