Quadtratic equations. 2.6

Question

The cost function for a certain company is
C = 50x + 400
and the revenue is given by
R = 100x − 0.5x2.
Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of x (production level) that will create a profit of $400.

 

Answer 1

C = 50x + 400
R = 100x - 0.5x²

 

P = R - C = 400

 

R - C = 100x - 0.5x² - 50x - 400 = 400

 

Put the equation in terms of zero:

-0.5x² + 50x - 800 = 0

Use quadratic equation to solve:

x = (-50 +/- sqrt(50²-4(-0.5)(-800))) / 2(-0.5)
x = (-50 +/- sqrt(2500-4(-0.5)(-800))) / -1
x = (-50 +/- sqrt(900)) / -1
x = (-50 +/- 30) / -1

x = 20, 80

Answer 2

Hi, Evelyn,

     C=50x + 400                and R= 100x- .5 x²         Also, Profit= Revenue —Cost        x= # of items produced

 Given:  Profit=$400

Use substitution to come up with the Quadratic Equation:  Profit= 100x- .5 x²  — (50x + 400 )   Simplify

 

400=100x- .5 x² — (50x + 400 )                    400=100x— .5x²—50x—400    

 

You need to get zero on one side and the rest of the equation on the right side in order to solve.  Subtract 400 from both sides of the equation:     400—400= 100x— .5x²—50x—400—400   Write Quadratic in standard form:

0=ax² +bx +c.

 

0= -.5x²+100x—50x—800              Note:   -.5=-5/10 = -1/2   easier to use

 

The parabola represented by the above equation crosses the x-axis at two places.  That is what you need to find.  You might want to use the Quadratic Formula to find the two values of "x."  You can also factor to find "x" as well.            Thank you,  Susan C.

 

Answer 3

Profit = Revenue - Costs

 

P = -5x² + 100x - (50x + 400)

P = -5x² + 100x - 50x - 400

P = -0.5x² + 50x - 400

 

You want P = $400

 

400 = -0.5x² + 50x - 400

0 = -0.5x² + 50x - 800

0 = -0.5(x² - 100x + 1600)

 

The quadratic factors:

 

0 = -0.5(x-20)(x-80)

 

x = 20 and 80

Answer 4

P = 100x - 0.5x² - (50x + 400)

P = 100x - 0.5x² - 50x - 400

P = -0.5x² + 50x - 400

 

400 = -0.5x² + 50x - 400

-0.5x² + 50x - 800 = 0

 

To make it easier to deal with, I am multiplying everything by -2.

 

x² - 100x + 1600 = 0

(x - 80)(x - 20)= 0

 

x - 80 = 0

x = 80

 

x - 20 = 0

x = 20

 

20 or 80