maths

(a)The sales revenue
=xp(x)
= -0.5x2+100x
Actual profit for selling the x boxes of contact lens
= xp(x)﹣C
=[-0.5x2+100x]﹣[(x/3)3+2x+8]
=-(x/3)3﹣0.5x2+98x﹣8
When selling 5 boxes, x=5
Profit
= -(5/3)3﹣0.5(5)2+98(5)﹣8
=464.87 dollars
When selling 6 boxes x=6
Profit
= -(6/3)3﹣0.5(6)2+98(6)﹣8
=554 dollars
The actual profit for selling the 6th boxes of contact lens
=554﹣464.87
= 89.13 dollars
(b)let f(x)=-(x/3)3﹣0.5x2+98x﹣8
f'(x) = -x2/9﹣x+98
The marginal profit function= -x2/9﹣x+98
Then approximately, the profit for selling the 6th boxes of contact lens
=f'(5)
= -25/9﹣5+98
=90.22 dollars