I. 1) x belongs to [0,10]
2) (JK) is parallel to (BC), therefore, angle AJM is equal to angle ABC (corresponding angles) and similarly angle AKM is equal to angle ACB. Since triangle ABC is isosceles, then angle ABC is equal to angle ACB, and thus angle AJM is equal to AKM. So triangle AJK is isosceles and [AM] height, therefore M midpoint of [JK]. So JM=MK.
3) (AM)/(AH)= (JM)/(BH)
(x)/(10)= (JM)/(6)
JM= (6x)/10
f(x)= JK*MH=2*JM*MH=2*(6x/10)*(10-x)=12x-6(x^2)/5
4) just develop it.
5) nothing new
6) max at (5,30)