Quadratic Equation Word Problem

Question

For a certain company, the cost for producing x items is 40x+300 and the revenue for selling x items is 80x−0.5x2.

 

The profit that the company makes is how much it takes in (revenue) minus how much it spends (cost). In economic models, one typically assumes that a company wants to maximize its profit, or at least wants to make a profit!

 

Part a: Set up an expression for the profit from producing and selling x items. We assume that the company sells all of the items that it produces. (Hint: it is a quadratic polynomial.)

 

Part b: Find two values of x  that will create a profit of $50.

 

The field below accepts a list of numbers or formulas separated by semicolons (e.g. 2;4;6 or x+1;x−1). The order of the list does not matter. To enter a−−√, type sqrt(a).

 

x= 

 

Part c: Is it possible for the company to make a profit of $2,500?

Answer 1

profit=40x-.5x^2-300

10, 70

make 10 or 70 to get 50 profit

50=40x -.5x^2 -300

divide by 5

10 = 8x -.1x^2 -60

.1x^2 -8x +70

multiply by 10

x^2 -80+700 = 0

(x-70)(x-10)

x=10 or 70

factor if possible

set each factor=0

use the quadratic formula, complete the square or graph

x = 80/2 + or - .5sqr(80^2 -4(700))

= 40+/- .5sqr3600= 40+/-30= 10 or 70

2500 =-.5x^2+40x-300

x^2 -80x +5600=0

discriminant = 6400-4(5600)<0

so no it's impossible to make a profit of 2500

max profit is when -x +40 = 0, when x =40

then profit = 800-300 = 500 = max