suppose that for a certain company, $c(x)=20x + 292000$ represents the total - cost function, and $r(x)=40x$…

suppose that for a certain company, $c(x)=20x + 292000$ represents the total - cost function, and $r(x)=40x$ represents the total - revenue function.\n(a) find the total - profit function.\n(b) find the break - even point.

suppose that for a certain company, $c(x)=20x + 292000$ represents the total - cost function, and $r(x)=40x$ represents the total - revenue function.\n(a) find the total - profit function.\n(b) find the break - even point.

Answer

Explanation:

Step1: Recall profit formula

Profit $P(x)=R(x)-C(x)$.

Step2: Substitute given functions

$P(x)=40x-(20x + 292000)=40x-20x-292000=20x - 292000$.

Step3: Recall break - even formula

At break - even, $R(x)=C(x)$.

Step4: Solve for $x$

$40x=20x + 292000$. Subtract $20x$ from both sides: $40x-20x=20x+292000 - 20x$, so $20x=292000$. Divide both sides by 20: $x=\frac{292000}{20}=14600$. When $x = 14600$, $R(14600)=40\times14600 = 584000$ and $C(14600)=20\times14600+292000=584000$.

Answer:

(a) $P(x)=20x - 292000$ (b) The break - even point is $(14600,584000)$