select the correct answer.

an entrepreneur estimates his total profit (total revenue minus total cost) for his proposed company as $p(x) = x^3 - 4x^2 + 5x - 20$, where $p$ is in hundreds of dollars and $x$ is number of years the company has been in business. in which year ($x$) will the entrepreneur break even?

- year 1
- year 2
- year 3
- year 4

Question

select the correct answer.

an entrepreneur estimates his total profit (total revenue minus total cost) for his proposed company as $p(x) = x^3 - 4x^2 + 5x - 20$, where $p$ is in hundreds of dollars and $x$ is number of years the company has been in business. in which year ($x$) will the entrepreneur break even?

- year 1
- year 2
- year 3
- year 4

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Break-Even Analysis", "Polynomial Factoring" ], "new_concepts": [], "current_concepts": [ "Break-Even Analysis", "Polynomial Factoring", "Cubic Functions" ] } ## Set the profit function to zero for break-even $$ p(x) = x^3 - 4x^2 + 5x - 20 = 0 $$ ## Factor the polynomial by grouping $$ \begin{aligned} x^2(x - 4) + 5(x - 4) &= 0 \ (x^2 + 5)(x - 4) &= 0 \end{aligned} $$ ## Solve for real roots of x $$ \begin{aligned} x - 4 = 0 &\implies x = 4 \ x^2 + 5 = 0 &\implies x = \pm i\sqrt{5} \quad (\text{no real solution}) \end{aligned} $$

year 1

year 2

year 3

year 4

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Break-Even Analysis" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Set the profit function to zero for break-even

Factor the polynomial by grouping

Solve for real roots of x

Answer: