College Algebra Word Problem

Question

A manufacturer of widgets has fixed costs of $610 per month, and the variable cost is $64 per thousand widgets (so it costs $64 to produce 1 thousand widgets). Let N be the number, in thousands, of widgets produced in a month.
(a) Find a formula for the manufacturer's total cost C as a function of N.
C(N) = ___
  

(b) The highest price p, in dollars per thousand widgets, at which N can be sold is given by the formula 
p = 75 − 0.02N.
 Using this, find a formula for the total revenue R as a function of N.
R(N) = ___
  

(c) Use your answers to parts (a) and (b) to find a formula for the profit P of this manufacturer as a function of N.
P(N) = ___
  

(d) Use your formula from part (c) to determine the two break-even points for this manufacturer. Assume that the manufacturer can produce at most 500 thousand widgets in a month. (Round your answers to two decimal places.)
___ thousand widgets per month (smaller value)
___ thousand widgets per month (larger value)

Answer 1

(a) 610 +64*N


(b) R(n) = n* p(n) = n* ( 75 - 0.02N) = 75n - 0.02n^2


(c) Profit = P(n) = 75n - 0.02n^2 - (610 + 64n) =
75n - 0.02n^2 - 610 - 64n

= 11n - 0.02n^2 - 610

= -0.02n^2 + 11n - 610


(d) P(n) = 0

-0.02n^2 + 11n - 610 = 0

0.02n^2 - 11n + 610 = 0

2n^2 - 1100n + 61000 = 0

n^2 - 550n + 30500 = 0

quadratic formula says:
n = 275 +or- 95*sqrt(5)

n = 487.426 or n = 62.5735

The latter solution satisifies the given constraints