Question 29084
i need help on the approach to solve the problem: the number eggs N in a female moth is a function of her abdominal width W, in millimeters, modeled by 
N=14W^3 - 17W^2 - 16W + 34 , for 1.5 < W < 3.5
what is the abdominal width when there are 211 eggs?
THAT IS WE HAVE TO FIND W SO THAT N BECOMES EQUAL TO 211.
N=14W^3 - 17W^2 - 16W + 34  =211
 for 1.5 < W < 3.5
14W^3 - 17W^2 - 16W + 34 - 211 =0
14W^3 - 17W^2 - 16W - 177 =0
14W^3 = 17W^2 + 16W + 177 .........I

SINCE  1.5 < W < 3.5,W IS POSITIVE..SO WE HAVE TO SEE THAT FOR POSITIVE W IN THE RANGE 1.5 < W < 3.5,L.H.S. = R.H.S.WE FIND THAT FOR POSITIVE W BOTH SIDES ARE POSITIVE AND FURTHER ALL TERMS ON R.H.S. ARE POSITIVE.SO WE TRY INTEGRAL VALUES FROM W =1 TO 2 TO 3 FIRST AND PROBE FOR INTEGRAL SOLUTION.SO WE GET 
W......................................1.........2...........3............
L.H.S.=14W^3=..........................14........112.........378..........
R.H.S.=17W^2 + 16W + 177 =.............210.......278.........378...........
SO W =3 MM 
NOTE THAT THIS IS THE STANDARD TRIAL AND ERROR APPROACH TO SOLVING THIRD OR HIGHHER DEGREE EQUATIONS.IF WE FAIL IN GETTING AN INTEGRAL VALUE,WE WILL ATLEAST KNOW THAT IT LIES BETWEEN 2 INTEGERS BY SEEING THAT AT ONE VALUE RHS>LHS ANMD AT THE NEXT VALUE LHS>RHS OR VICE VERSA..WHICH TELLS US THAT THE ANSWER IS BETWEEN THOSE 2 INTEGERS.HERE AS W INCREASES , WE FIND LHS IS CATCHING UP WITH RHS.WE GOT AN INTEGRAL ANSWER IN THIS CASE.
OTHER METHOD IS TO DRAW GRAPH OF LHS AND RHS IN ONE SHEET AND SEE WHERE THEY INTERSECT...SEE BELOW...FOR GRAPHICAL METHOD...

{{{ graph( 600, 600, -3, 4, -50, 1000, 14*x^3, 17*x^2+16*x+177) }}}
WE FIND THAT THE 2 CURVES MEET AT X=3 OR W=3..IS THE SOLUTION.