Conversion of Solids PYQs
Advanced Practice (Legacy Board Patterns)
Topic Overview
When a solid is melted and recast into another shape, its Volume remains constant. This principle is used to find the number of objects or new dimensions. Note: This topic is technically legacy but remains excellent for volume mastery.
Q1
2025
00:00
Three metallic solid spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.
Volume of 3 spheres $= (4/3)\pi(6^3 + 8^3 + 10^3) = (4/3)\pi(216 + 512 + 1000) = (4/3)\pi(1728)$.
Let new radius be $R$. $(4/3)\pi R^3 = (4/3)\pi(1728)$.
$R^3 = 1728 \Rightarrow R = 12$ cm.
Ans: 12 cm
Q2
2020
00:00
A solid sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed.
Volume of sphere $= (4/3)\pi(10.5)^3$.
Volume of one cone $= (1/3)\pi(3.5)^2(3) = \pi(3.5)^2$.
Number of cones $= 126$.
Ans: 126
Q3
2019
00:00
A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness (diameter) of the wire.
Volume of rod $= \pi(0.5)^2 \times 8 = 2\pi$ cm³.
Volume of wire $= \pi r^2 \times 1800$ cm³.
$\pi r^2 \times 1800 = 2\pi \Rightarrow r^2 = 1/900 \Rightarrow r = 1/30$ cm.
Diameter $= 2/30 = 1/15$ cm $\approx 0.67$ mm.
Ans: 1/15 cm
Q4
2018
00:00
Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm of standing water is needed?
Volume in 30 mins $= 6 \times 1.5 \times 10,000 \times (30/60) = 45,000$ m³.
Area to irrigate $= 45,000 / 0.08 = 562,500$ m².
Ans: 56.25 hectares
Three metallic solid spheres of radii 6 cm, 8 cm and 10 cm, respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.
Volume of 3 spheres $= (4/3)\pi(6^3 + 8^3 + 10^3) = (4/3)\pi(216 + 512 + 1000) = (4/3)\pi(1728)$.
Let new radius be $R$. $(4/3)\pi R^3 = (4/3)\pi(1728)$.
$R^3 = 1728 \Rightarrow R = 12$ cm.
Ans: 12 cm
Q2
2020
00:00
A solid sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed.
Volume of sphere $= (4/3)\pi(10.5)^3$.
Volume of one cone $= (1/3)\pi(3.5)^2(3) = \pi(3.5)^2$.
Number of cones $= 126$.
Ans: 126
Q3
2019
00:00
A copper rod of diameter 1 cm and length 8 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness (diameter) of the wire.
Volume of rod $= \pi(0.5)^2 \times 8 = 2\pi$ cm³.
Volume of wire $= \pi r^2 \times 1800$ cm³.
$\pi r^2 \times 1800 = 2\pi \Rightarrow r^2 = 1/900 \Rightarrow r = 1/30$ cm.
Diameter $= 2/30 = 1/15$ cm $\approx 0.67$ mm.
Ans: 1/15 cm
Q4
2018
00:00
Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm of standing water is needed?
Volume in 30 mins $= 6 \times 1.5 \times 10,000 \times (30/60) = 45,000$ m³.
Area to irrigate $= 45,000 / 0.08 = 562,500$ m².
Ans: 56.25 hectares