Mensuration — Part 2: Cost problems, melting and recasting, combinations of solids: visual lesson, ICSE Class 10 Mathematics

Lesson

Cost, melting and combinations

In part 2 the formulas go to work: how much will it cost to paint or fill a solid, what comes out when a solid is melted and recast, and what happens when solids are joined.

In this lesson you will

Find the cost of painting a surface or filling a container from its area or volume
Find the volume of the material in a hollow solid from its outer and inner volumes
Use equality of volume to find the number or dimensions of solids formed by melting and recasting
Find the surface area and volume of a solid made of a cylinder, a cone and a hemisphere

Melting and recasting changes the shape, not the volume.

Concept

What you will be able to do

Find the cost of painting, polishing or filling
Find the metal in a hollow solid: inner and outer volume
Use equal volumes for melting and recasting
Handle combinations of a cylinder, cone and hemisphere

Part 2 on one card.
Type of problem	Rule	Typical mistake
Cost of painting or plating	area $\times$ rate per square unit	using volume, or the wrong surfaces
Cost of filling a tank	litres $\times$ rate per litre	forgetting to convert cubic metres to litres
Hollow solid or shell	outer volume $-$ inner volume	using the thickness as the radius
Melting and recasting	volume before $=$ volume after	equating surface areas
Combination of solids	areas of the outside surfaces; volumes added	counting the joined faces

Concept

Units first

Costs are given per square unit or per litre. Convert before you multiply, so that the units of the quantity and the rate match.

a tank in cubic metres is filled in litres: multiply by $1000$
a radius in cm and a rate per m $^2$ need one common unit

Convert before you multiply by a rate.
Conversion	Value
1 cubic metre	1000 litres
1 litre	1000 cubic centimetres
1 metre	100 centimetres
1 square metre	10000 square centimetres
1 kilolitre	1 cubic metre

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