Question

if a cone of radius 10 cm is divided into two parts by drawing a plane through the midpoint of its axis, parallel to its base find the ratio of volume of two parts

Answer 1

Let a cone of radius r = 10 cm 
height = h cm

a/q,
volume of complete cone = πr²h/3
volume of part (i)  = π (r')² h/2
Volume of  part (ii) = total volume - Volume of  part(i)
                              = πr²h - π(r')² h/2
                              = πh{r² - (r')²/2}

Now,                            
l²  = r² + h²
⇒ r² = l² - h² -------------(1)

and 
(l/2)² = (r')² + (h/2)²
(r')² = (l/2)² - (h/2)² 
$r ' = \sqrt{(l/2 )^{2} - (h/2 )^{2} } \\ r'= \sqrt{ \frac{ l^{2}- h^{2} }{4}} \\ r'= \frac{1}{2} \sqrt{ l^{2}- h^{2} }$
r' = r/2
⇒ (r')² =  r²/4
Now, 
Volume of part (ii) = πh[r² - (r²/4)*(1/2)]
                              = πh(r² - r²/8)
                              = 7πhr²/8
And,
Volume of Part (i) = π(r/2)² h = πr²h/4
Ratio of volume of two parts = volume of part (i) ÷ volume of part (ii)
                                              =  πr²h/4 ÷  7πr²h/8
                                              = 2/7 =  2:7