Answer:
MN = 7 cm
Step-by-step explanation:
Well the volume we can separate into pieces
first the ABCDEFGH (? you know this cuboid)
which has a volume of
[tex]V = AB\cdot BC\cdot CH\\\\V = 3*6*(3+JY+4)[/tex]
if you ask where does the 4 come from
it comes from LP = KB = YC and JY + YC =JC
so
[tex]V = 18(7+MX)[/tex]
this is because JY = MX
the other volume is of BCNPKLXY (hope you understand which one)
and the volume is
[tex]V = BP*PN*NX\\\\V=8*6*4\\\\V = 192[/tex]
and the last one is KYXLMJ (the top "triangle")
the volume is
[tex]V = \frac{LX*XM}{2}*LK\\\\V = \frac{6*MX}{2}*8\\\\V= 6*4*MX\\\\V=24MX[/tex]
so now add them up
and has to equal 444
[tex]V_T = 18(7+MX)+192+24MX\\\\V_T = 126+18MX+192+24MX\\\\V_T = 318+42MX\\\\444=318+42MX\\\\444-318=42MX\\\\126=42MX\\\\MX = 3[/tex]
and finally we know that MN = MX + XN = 3+4 = 7
so here is our answer MN = 7 cm
a couple of notes
[tex]1.\:\angle LXM = 90\\\\2.\:\angle KYX = 90[/tex]
3. the units don't matter due to the fact that everything is in cm
4. see the picture below to get a better idea of which prism I am talking of
5. tell me if you have questions :D
