A cube is placed inside a cone of radius 20 cm and height
10 cm, one of its face being on the base of the cone and
vertices of opposite face touching the cone. What is the
length (in cm) of side of the cube?

[20 - x/√2]/x = 20/10

20 - x/√2 = 2x

20 = x[2 +1/√2]

20√2= x [2√2+1]

x= 20√2/[1+2√2]

The radius of base of a solid cylinder is 7 cm and its height
is 21 cm. It is melted and converted into small bullets.
Each bullet is of same size. Each bullet consisted of two
parts viz. a cylinder and a hemisphere on one of its base.
The total height of bullet is 3.5 cm and radius of base is 2.1
cm. Approximately how many complete bullets can be
obtained?

7*7*21/(2.1*2.1*2.8
.
250/3 = 83 ??

49*21 = n( 2.1² *1.4 + 2/3 *2.1³)
490/2.1 = n( 2.8)
1000/12 =n
83 = n

Let the equal sides are a
Unequal side b
Perimeter 200
2a+b=200

Triangles inequality
2a>b
Put b from the above
2a>200-2a
A>50
A Will be maximum integer at b=2
2a=200-2
A=99
So 50<a<99
49 values

An equliteral triangle inscribed in a rectangle

Remember ,a useful result when this equliteral triangle inscribed in a square it always inclines at 75° with side {√6-√2}x

A right angle ,a square inside, radius of same colour are same find the relationship Between the radius of these circles.

Also prove that
Area of triangle AEF=a quadliratial EFGH

A very useful and important results