Question
Two solid cylinders, one with diameter $60 \ cm$ and height $30 \ cm$ and the other with radius $30 \ cm$ and height $60 \ cm, $aremetledandrecastedinto a third solid cylinder of height $10 \ cm$. Find the diameter of the cylinder formed.
✓
Answer
For cylinder 1,
Height $= h_1 = 30 cm$
Radius $=r_1=\frac{60}{2}=30 cm$
Volume $= V _1=\pi r _1^2 h _1=\pi \times 30 \times 30 \times 30=27000 \pi cm ^3$
For cylinder 2 ,
Height $= h_2 = 60 cm$
Radius $= r_2 = 30 cm$
Volume $= V _2=\pi r _2^2 h _2=\pi \times 30 \times 30 \times 60=54000 \pi cm ^3$
Let r be the radius of the third cylinder.
Height = h = 10 cm
Volume $= V = \pi r^2h = \pi r^2 \times 10$
Now,
$V = V_1 + V_2$
$\Rightarrow \pi r^2 \times 10 = 27000 \pi + 54000 \pi$
$\Rightarrow \pi r^2 \times 10 = 81000 \pi$
$\Rightarrow r^2 = 8100$
$\Rightarrow r = 90$
$\Rightarrow $ Diameter $= 2r = 180 cm$
Height $= h_1 = 30 cm$
Radius $=r_1=\frac{60}{2}=30 cm$
Volume $= V _1=\pi r _1^2 h _1=\pi \times 30 \times 30 \times 30=27000 \pi cm ^3$
For cylinder 2 ,
Height $= h_2 = 60 cm$
Radius $= r_2 = 30 cm$
Volume $= V _2=\pi r _2^2 h _2=\pi \times 30 \times 30 \times 60=54000 \pi cm ^3$
Let r be the radius of the third cylinder.
Height = h = 10 cm
Volume $= V = \pi r^2h = \pi r^2 \times 10$
Now,
$V = V_1 + V_2$
$\Rightarrow \pi r^2 \times 10 = 27000 \pi + 54000 \pi$
$\Rightarrow \pi r^2 \times 10 = 81000 \pi$
$\Rightarrow r^2 = 8100$
$\Rightarrow r = 90$
$\Rightarrow $ Diameter $= 2r = 180 cm$
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