For a certain company, the cost of producing x items is 40x + 300 and the revenue for selling x items is 80x - 0.5x^2. Find two values of x that will create a profit of $300.

Answer 1

To find the values of \( x \) that will create a profit of $300, we need to set up the profit equation and solve for \( x \).

First, let's define the profit equation. Profit \( P \) is given by the revenue \( R \) minus the cost \( C \):

\[ P = R - C \]

Given:

- The cost \( C \) of producing \( x \) items is \( 40x + 300 \).

- The revenue \( R \) for selling \( x \) items is \( 80x - 0.5x^2 \).

The profit equation is:

\[ P = (80x - 0.5x^2) - (40x + 300) \]

Simplify the equation:

\[ P = 80x - 0.5x^2 - 40x - 300 \]

\[ P = 40x - 0.5x^2 - 300 \]

We are given that the profit \( P \) is $300:

\[ 40x - 0.5x^2 - 300 = 300 \]

Rearrange the equation to set it to 0:

\[ 40x - 0.5x^2 - 300 - 300 = 0 \]

\[ 40x - 0.5x^2 - 600 = 0 \]

Multiply the entire equation by 2 to eliminate the fraction:

\[ 80x - x^2 - 1200 = 0 \]

Rewrite it as a standard quadratic equation:

\[ -x^2 + 80x - 1200 = 0 \]

\[ x^2 - 80x + 1200 = 0 \]

We can solve this quadratic equation using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = -80 \), and \( c = 1200 \):

\[ x = \frac{-(-80) \pm \sqrt{(-80)^2 - 4 \cdot 1 \cdot 1200}}{2 \cdot 1} \]

\[ x = \frac{80 \pm \sqrt{6400 - 4800}}{2} \]

\[ x = \frac{80 \pm \sqrt{1600}}{2} \]

\[ x = \frac{80 \pm 40}{2} \]

This gives us two solutions:

\[ x = \frac{80 + 40}{2} = \frac{120}{2} = 60 \]

\[ x = \frac{80 - 40}{2} = \frac{40}{2} = 20 \]

Therefore, the two values of \( x \) that will create a profit of $300 are:

\[ \boxed{20 \text{ and } 60} \]

Answer 2

P=r=c = -x^2/2+40x - 300=300

-.5x^2 +40x -600=0

x=40+/- sqr(1600-1200)

= 20 or 60

check 20

P= -20^2/2 +80(20)- 40(20)-300=1400-1100=300

check 60

P=-60^2/2+80(60)-40(60)-300=300

or

-x^2/2 +40x-600=0

multiply by -2 and factor

x^2-80x +1200=0

(x-20)(x-60)=0

x=20, 60

or graph the quadratic and y=300

and see where they intersect

use desmos online graphing calculator or TI83

intersections are (20, 300) and (60,300)