Question

The height of a right circular cone is trisected by two planes drawn parallel to base.Show that the volumes of the three portions starting from the top are in the ratio 1 : 7 : 19.

Answer 1

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Answer 2

Proved that volumes of the three portions are in the ratio 1 : 7 : 19.

Given

To show that volumes of three portions are in the ratio 1 : 7 : 19

Height of a circular cone is trisected by two planes which is drawn parallel to base.                                                                                                

From the figure,

                         AQ = QY = YD = h

                                             QR = r

                         PQR and XYZ are the two planes intersects ABDC.

In ΔAQR and ΔAYZ,

       ∠A = ∠A  ( ∠A is common  angle )

By AA ( Angle-Angle ) similarity,

       ΔAQR = ΔAY        

       AQ/AY = QR/YZ    ( Corresponding sides of similar triangle's (CSST) )

           h/2h = r/YZ        ( where, AY = 2h )

            YZ = 2r

Similarly, ΔAQR ≅ ΔADC,      

         AQ/AD = QR/DC   ( Corresponding sides of similar triangle's (CSST) )

           h/3h = r/DC          ( where, AD = 3h )

             DC = 3r

Let volume of cone APR, volume of frustum PXZR and volume of frustum XBCZ are in the ratio of $v_{1} : v_{2} : v_{3}$

To find volume of cone, APR ( $v_1$ ) :

            Volume of cone = 1/3 × πr²h   -----> ( 1 )

To find volume of frustum PXZR ( $v_2$ ) :

            Volume of frustum = 1/3 × πh [ (2r)²+2r×r+r² ]

                                            = 1/3 × πh × [ 4r²+2r²+r² ]

                                            = 1/3 × πh × 7r²

                                            = 7 × 1/3 × πr²h  -----> ( 2 )

To find volume of frustum XBCZ ( $v_3$ ) :

            Volume of frustum = 1/3 × πh [ (3r)²+3r×2r+(2r)² ]

                                            = 1/3 × πh [ 9r²+6r²+4r² ]

                                            = 1/3 × πh × 19r²

                                            = 19 × 1/3 × πr²h  -----> ( 3 )

Hence, the ratio of $v_1 : v_2 : v_3$ =  1/3 × πr²h : 7 × 1/3 × πr²h : 19 × 1/3 × πr²h

                                                = 1 : 7 : 19.

Therefore, the volumes of the three portions are in the ratio 1 : 7 : 19.

Hence proved.

To learn more...

1. brainly.in/question/5181951

2. brainly.in/question/7115497