Question : A solid toy is in the form of a hemisphere surmounted by a right circular cone. Ratio of the radius of the cone to its slant height is 3:5. If the volume of the toy is 240π cm³, then find the total height of the toy.
CBSE 2024 (July Session)
Solution :
For Right Circular Cone
r:l=3:5
Let
r=3x
l=5x
Height of the cone
h=√[l²-r²]
h=√[(5x)²-(3x)²]
h=√[25x²-9x²]
h=√[16x²]
h=4x
Also the radius of hemisphere would be equal to that of cone.
Volume of the toy= Volume of cone + Volume
f hemisphere
240π = (1/3)πr²h + (2/3)πr³
240π = (1/3)πr²[h+2r]
240=(1/3)(3x)²[4x+2(3x)]
240=3x²[4x+6x]
240=3x²[10x]
240=30x³
240=30x³
240/30=x³
8=x³
x³=2³
x=2
Total height of the toy
= h+r
= 4x+3x
=7x
=7(2) [∵x=2]
=14 cm