Qyadratic Function

Question

The cost of producing computer software is a function of the number of hours worked by the employees. If the cost function can be approximated by
 C(x)= 0.04x²-20x+6000      Where 200≤x≤1000

a) What would be the cost if the employees worked 800 hours?

b) Find the number of hours the employees should work in order to minimize the cost. What would the minimum cost be?

Answer 1

C(800)  = 0.04(800)^2 - 20*800 + 6000 = 15600

 

 

 

Need to find the AVERAGE of the zeros.

 

  0 = 0.04x^2 - 20x + 6000

 

 0 = 0.04(x^2 - 500x   + 62500  ) + 6000 - 2500<--- completing the square

 

 0 = 0.04( x - 250)^2 + 3500

 

 The vertex is (h,k) = (250,3500)

 

250 hours will minimize the cost to $3500

 

Answer 2

(a)  C(800) = 0.04*800² - 20*800 + 6000 = 25,600 -16000 + 6000 = 15,600 hours

 

(b)  Set the equation equal to 0, to get the 0's:  

 

First, get rid of the decimal number, by multiplying the equation by 100, to give us:

 

4x² - 2000x + 600000 = 0

 

To simplify this, divide this out by 4, to give us:

 

x² - 500x + 150000 = 0

 

Factor this, to get the 0's:

 

Reminder:  take the 2nd sign, it's a positive, which means that both of the signs are alike

Reminder, Part 2:  the first sign, given the second sign, means that both numbers are a negative;

Reminder, Part 3:  two numbers combined equals '-500', multiplied together give us '150,000'; which gives us:  not so easy.  Easier:  quadratic instead, to get:

 

x = 250 + 5i (35)^½ and x=250 - 5i (35)^½    (remember, to the ½ power is the same thing as the square root)

 

Although this might look nifty, don't stop here!  Test with both of the x-valued:

 

[(250 + 5i (35)^½]² - 500 [(250 + 5i (35)^½ + 150,000 = ?   (you can do it - you can solve this now)!

 

[(250 - 5i (35^½]² - 500 [(250 - 5i (35)^½ + 150,000 = ?   (you can do it - you can solve this now)!

 

So, which of these is correct?

 

 

Answer 3

a)

 

Evaluate C(800).

 

 

b)

 

Use the formula

 

h = -b / (2a)

 

where:

a = 0.04

b = -20

 

to find the number of hours for minimum cost.

 

Then evaluate C(h) to find the minimum cost.