Question

the height of the cone is 120 CM a small cone is off at the top by a plane parallel to the base and volume is 1/64 the volume of the original cone the height the base at which the section is made is​

Answer 1

Answer:

The height from the base at which the section is made  will be 90 cm

Step-by-step explanation:

Since the smaller cone and larger cones are similar to each other

hence their radius and heights are proportional to each other

=> R/r = H/h = constant = k

where R = radius of larger cone

r = radius of smaller cone

H = height of big cone

h = height of small cone

Given their volume are in the ratio of 64 : 1

=> V/v = 64

=> (πR²H/3)/(πr²h/3) = 64

=> R²/r² x H/h = 64

=> k² x k = 64

=> k³ = 64

=> k = 4

Hence H/h = k = 4

=> 120/h = 4

=> h = 30 cm

Hence the height from the base at which the section is made = 120 - 30

= 90 cm

Answer 2

answer : 90cm from the base of which the section is made.

height of the cone , h = 120cm

Let a small cone is cut off at the top by a plane parallel to the base.

length of small cone is l and radius of small cone is x.

a/c to question,

volume of small cone = 1/64 × volume of original cone

or, 1/3 πx²l = 1/64 × πR²h

or, x²l = 1/64 × R² × 120

or, x²l = 15R²/8 ..........(1)

see figure, from ∆ABC and ∆AED

$\angle{ABC}=\angle{AED}$ [corresponding angles ]

$\angle{ACB}=\angle{ADE}$ [ corresponding angles ]

so, $\Delta ABC\sim\Delta AED$

$\therefore\frac{AC}{AD}=\frac{CB}{DE}$

$\frac{l}{h}=\frac{x}{R}$

or, lR = hx = 120x

or, l = 120x/R .......(2)

from equations (1) and (2),

x² × (120x/R) = 15R²/8

or, 120x³ = 15R³/8

or, 64x³ = R³

or, x/R = 1/4

hence, x = R/4 , put it in equation (2),

l = 120 × (R/4)/R = 30cm

hence, (120cm - 30cm) = 90cm from the base of which the section is made.