Question

6.

For this problem we will use the fact that:

?????? = ??????? − ????

? ? = ? ? − ?(?)

A company produces and sells copies of an accounting program for home

computers. The total weekly cost (in dollars) to produce x copies of the program

is ? ? = 8? + 500, and the weekly revenue for selling all x copies of the program

is ? ? = 35? − 0.1?).

a. Find a function, ?(?), for the profit of producing and selling x copies.

b. How many programs must be sold each week for the profit to be $1200?

c. How many programs do they need to sell to maximize their profit? What is

the maximum profit?

Answer 1

Sol.-

cost function C(x)= 8x+500 and Revenue function R(x)=35x-0.1x2

(a) As we know that Profit function P(x) is given by

P(x)= R(x)-C(x)

P(x)=( 35x-0.1x2 ) -(8x+500)

P(x)= 27x-0.1x2 -500 -----(1)

(b) for profit P(x)= $ 1200 we have

27x-0.1x2 -500=1200   [ by using eq.(1)]

-0.1x2 +27x -500-1200=0

-0.1x2 +27x -1700=0

x2 -270x-17000=0  [ dividing  by -0.1 ]

So for profit of $1200 we must sell either 100 or 170 programs each week.

(c) Now on differentiate eq.(1)w.r.t. x we get

P'(x)=27-2(0.1)x-0

P'(x)=27-0.2x -------(2)

for maxima put P'(x)=0

27-0.2x=0

0.2x=27

x=27/0.2

x=135

again differentiating eq.(2) w.r.t. x we get

P"(x)=0-0.2(1)= -0.2 < 0

maxima occurs at x=135

So, maximum profit occurs at x=135 and maximum profit is P(135)=27(135)-0.1(135)2 -500

P(135)= 3645-1822.5-500

P(135)=$1322.5

Hence maximum profit occurs on selling 135 programs and maximum profit is $1322.5.