Area and volume ratios, removed cones and nested shapes
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Area and volume ratios, removed cones and nested shapes
Optional extension: Area and volume ratios, removed cones and nested shapes
Example 1
A cone of volume 135cm3 has a smaller similar cone removed from its top by a cut parallel to the base. The remaining volume is 130cm3. If the original height was 9cm, find the removed cone's height.
Cone cut parallel to baseDiagram not to scaleShow worked answer
Answer: 3
The removed volume is 5cm3
its volume ratio to the original is 1:27.
The height ratio is 1:3.
Divide 9 by 3.
Example 2
In triangle ABC, DE is parallel to BC with D on AB and E on AC. AD=3cm and DB=18cm. The area of trapezium DBCE is 432cm2. Find the area of triangle ADE.
Similar trianglesDiagram not to scaleShow worked answer
Answer: 9
ABAD=71.
Similar triangle areas therefore have ratio 1:49.
The trapezium is 48 times the smaller triangle
divide 432 by 48.
Example 3
A cone of volume 256cm3 has a smaller similar cone removed from its top by a cut parallel to the base. The remaining volume is 252cm3. If the original height was 16cm, find the removed cone's height.
Cone cut parallel to baseDiagram not to scaleShow worked answer
Answer: 4
The removed volume is 4cm3
its volume ratio to the original is 1:64.
The height ratio is 1:4.
Divide 16 by 4.
Put it into practice.
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