Let's break down this problem step-by-step.
1. Revenue and Cost Expressions:
The revenue expression for selling [tex]\(x\)[/tex] video game systems is given by:
[tex]\[
\text{Revenue} = 5x^2 + 2x - 80
\][/tex]
The cost expression for producing [tex]\(x\)[/tex] video game systems is given by:
[tex]\[
\text{Cost} = 5x^2 - x + 100
\][/tex]
2. Profit Expression:
Profit is defined as the difference between revenue and cost. Thus, the profit expression [tex]\(P(x)\)[/tex] is:
[tex]\[
P(x) = \text{Revenue} - \text{Cost}
\][/tex]
Substituting the given expressions for revenue and cost, we get:
[tex]\[
P(x) = (5x^2 + 2x - 80) - (5x^2 - x + 100)
\][/tex]
3. Simplifying the Profit Expression:
To simplify [tex]\(P(x)\)[/tex], let's distribute and combine like terms:
[tex]\[
P(x) = 5x^2 + 2x - 80 - 5x^2 + x - 100
\][/tex]
Combine like terms:
[tex]\[
P(x) = (5x^2 - 5x^2) + (2x + x) + (-80 - 100)
\][/tex]
[tex]\[
P(x) = 0 + 3x - 180
\][/tex]
[tex]\[
P(x) = 3x - 180
\][/tex]
4. Profit Expression:
So, the profit expression can be modeled by the polynomial:
[tex]\[
P(x) = 3x - 180
\][/tex]
5. Calculating Profit for 1,000 Units:
To find the profit for selling 1,000 video game systems, substitute [tex]\(x = 1000\)[/tex] into the profit expression:
[tex]\[
P(1000) = 3(1000) - 180
\][/tex]
[tex]\[
P(1000) = 3000 - 180
\][/tex]
[tex]\[
P(1000) = 2820
\][/tex]
6. Conclusion:
Therefore, the profit expression is:
[tex]\[
P(x) = 3x - 180
\][/tex]
If 1,000 video game systems are sold, the company's profit is:
[tex]\[
\$2820
\][/tex]
