Question

A cone was cut parallel to its base into 2 pieces in such a way that the ratio of volume of smaller cone formed and other part is 64:61 .if the height of the cone is 40.5cm then find the height from the base at which the cut was made.

Answer 1

Given :

A cone was cut parallel to its base into 2 pieces in such a way that the ratio of volume of smaller cone formed and other part is 64 : 61

The height of the cone = H = 40.5 cm

To Find :

The height from the base at which the cut was made

Solution :

Total height of cone = H = 40.5 cm

Let , The height from the base at which the cut was made = h

∵ Cone was cut parallel to its base into 2 pieces

Height of upper cone = H - h =  (40.5 - h) cm

Height of lower cone = h  cm

∵  The volume of cone = $\dfrac{1}{3}$ × π × radius² × height

                                      = $\dfrac{1}{3}$ × π × r² × h

Volume of upper cone =   $\dfrac{1}{3}$ × π × r² × (40.5 - h)

Volume of lower cone =   $\dfrac{1}{3}$ × π × r² × ( h)

∵ The ratio of volume of upper cone to volume of lower cone = 64 : 61

i.e   $\dfrac{volume of upper cone}{volume of lower cone}$  = $\dfrac{64}{61}$

Or,    $\dfrac{\dfrac{1}{3}\pi r^{2} (40.5-h) }{\dfrac{1}{3}\pi r^{2} h}$  = $\dfrac{64}{61}$

Or,   61 ( 40.5 - h ) = 64 h

Or,   2430 - 61 h = 64 h

Or,  64 h + 61 h = 2430

Or,            125 h = 2430

∴                     h = $\dfrac{2430}{125}$

i.e                   h = 19.44 cm

Height = h = 19.44 cm

Hence, The height from the base at which the cone cut was made is 19.44 cm  Answer