Prove cota+cotb+cotc > = square root of 3

Given cotA + cotB + cotC = sqrt3
to prove triangle ABC is equilateral 
we prove this by assuming ABC to be equialteral and establishing the truth 
of the statement cotA + cotB + cotC =sqrt3 

since ABC is equialteral angleA=angleB=angleC=60 degrees 
cotA=cotB=CotC = cot60= 1/sqrt3 
therefore cotA + cotB + cotC = 1/sqrt3 + 1/sqrt3 + 1/sqrt3 
=3/sqrt3 
=sqrt3 which is equal to the RHS ( right hand side) of the expression 
hence our assumption that ABC is equilateral is true

Three teachers are needed for 70 children, that is:
3 = 70  equation 1.
 For 14,700 children, Y number of teachers are needed, thus, we have
Y = 14,700equation 2.
3 = 70
Y = 14, 700.
Cross multiplying we give us,
3 * 14, 700/70 * Y.
44,100/70Y
Y = 630.
Thus, 630 teachers are needed to teach 14,700 children.
The district already have 612 teachers, so they need 630 - 612 = 18. 
So, the district needs 18 more teachers.