Truck Tank Mystery

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Spot the Shapes

Observe this fuel truck on the road.

Its tank combines a straight cylinder and two perfectly matching hemispheres.

Recognising this composite solid helps us calculate how much metal covers it.

Key Points:

  • Real-world example: fuel truck tank
  • Shapes recognised: 1 cylinder + 2 hemispheres
  • Aim: find total outer surface area

What is a Composite Solid?

Composite Solid

A three-dimensional figure formed by joining two or more basic solids so that some of their faces become internal and disappear from view.

Surface Area Game-Plan

Follow these checkpoints to avoid double-counting and reveal every exposed face.

1

Decompose & Sketch

Draw the composite figure and label each basic solid clearly.

2

Highlight Exposed Faces

Shade only faces visible after joining; ignore internal contact areas.

3

Write Area Formulae

List the correct formula for each highlighted face with its dimensions.

4

Adjust for Overlaps

Subtract hidden or common areas, or add extra caps, to prevent double-counting.

5

Sum & State Units

Add all adjusted areas and write the answer with correct units, e.g., cm2.

Pro Tip:

Write a short note explaining every subtraction—examiners award marks for the reasoning.

Capsule Formula

\[TSA = 2\pi r h + 4\pi r^{2} = 2\pi r (h + 2r)\]

Add the curved area of the cylinder \(2\pi r h\) to the surface of two hemispheres \(4\pi r^{2}\).

Variable Definitions

\(r\) Common radius of cylinder and hemispheres
\(h\) Height of cylindrical part only
\(TSA\) Total surface area of the capsule

Applications

Medicine Capsules

Designing gelatin shells for pills uses this formula.

Storage Tanks

LPG and compressed-air vessels often have hemispherical ends.

Submersibles

Pressure-resistant crew modules combine cylindrical and hemispherical sections.

Play with Dimensions

Capsule Surface Explorer

Drag the sliders for radius \(r\) and height \(h\). Watch the capsule sketch, numbers, and bar chart change instantly. Observe how boosting \(r\) raises both hemispheres and cylinder area, while increasing \(h\) affects only the cylinder. Use these visuals to predict the total surface area.

Challenge: adjust \(r\) and \(h\) until the hemispheres make 60 % of the total surface area.

Worked Example

Cone-hemisphere toy

Cone-hemisphere toy

Example: Cone + Hemisphere Surface Area

A spinning top is formed by fixing a cone of radius 3 cm and slant height 5 cm on a hemisphere of the same radius.

Find the curved surface area to be polished using \( \text{CSA} = \pi r l + 2\pi r^{2} \). Apply the procedure for a cone-hemisphere combination.

Key Steps:

  • Identify common radius \( r = 3\text{ cm} \).
  • Cone area \( = \pi r l = \pi \times 3 \times 5 \,\text{cm}^2 \).
  • Hemisphere area \( = 2\pi r^{2} = 2\pi \times 3^{2} \,\text{cm}^2 \).
  • Add the two results for total curved surface area.

Multiple Choice Question

Question

A solid toy is made by mounting a hemisphere of radius 7 cm on top of a right circular cylinder of the same radius and height 15 cm. You need to paint all the outer surfaces except the circular base of the cylinder. Which areas must be added to get the required surface area?

1
Curved surface area of cylinder + curved surface area of hemisphere
2
Curved surface area of cylinder + total surface area of hemisphere
3
Total surface area of cylinder + total surface area of hemisphere
4
Curved surface area of cylinder only

Hint:

Ignore any circular face that gets covered or stays unpainted.

Key Takeaways

Recap: Surface area of a composite solid = sum of all visible faces.

First mark faces that disappear where solids join.

Apply standard formulas for each visible part (cube, cylinder, sphere, …).

Add the areas and state the answer with correct units.

Thank You!

We hope you found this lesson informative and engaging.