# A cylinder and a cone have equal bases. The height of the cylinder is 3 cm and the area of its base is 100 cm2 .

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A cylinder and a cone have equal bases. The height of the cylinder is 3 cm and the area of its base is 100 cm2 .The cone is placed upon the cylinder. Volume of the solid figure so formed is 500 cm3 . Find the total height of the figure

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Height of the cylinder, h = 3 cm
Let the radius of the cylinder be r cm and the height of the cone be H cm.
Area of the base of cylinder = 100 cm2

$\therefore \pi r^2 = 100$   .....(1)
The volume of the solid figure = 500 cm3
∴ Volume of the cylinder + Volume of the cone = 500 cm

$\Rightarrow \pi r^2 h + \frac{1}{3}\pi r^2 H = 500$
$\Rightarrow \pi r^2 \left( h + \frac{H}{3} \right) = 500$
$\Rightarrow 100\left( 3 + \frac{H}{3} \right) = 500 \left[ \text{ Using } \left( 1 \right) \right]$
$\Rightarrow 3 + \frac{H}{3} = \frac{500}{100} = 5$
$\Rightarrow \frac{H}{3} = 5 - 3 = 2$
$\Rightarrow H = 6$ cm

∴ Total height of the figure = h + H = 3 + 6 = 9 cm

Thus, the total height of the figure is 9 cm.