a painting company is adding painters caps to its product line. for the first year, the fixed costs for…

a painting company is adding painters caps to its product line. for the first year, the fixed costs for setting up production are $25,038. the variable costs for producing a dozen caps are $6. the revenue on each dozen caps will be $24. answer parts a through e. b) find the total - revenue function r(x) from the sale of x dozen caps. the total - revenue function is r(x)=24x. (simplify your answer. do not include the $ symbol in your answer.) c) find the total profit p(x) from the production and sale of x dozen caps. the total - profit function is p(x)=18x - 25038. (simplify your answer. do not include the $ symbol in your answer.) d) find the profit or loss from the production and sale of 3000 dozen caps and 1000 dozen caps. the production and sale of 3000 dozen caps will result in a profit of $28962. the production and sale of 1000 dozen caps will result in a loss of $7038. e) find the break - even point. the break - even point is (type an ordered pair using integers or decimals. do not include the $ symbol in your answer.)

a painting company is adding painters caps to its product line. for the first year, the fixed costs for setting up production are $25,038. the variable costs for producing a dozen caps are $6. the revenue on each dozen caps will be $24. answer parts a through e. b) find the total - revenue function r(x) from the sale of x dozen caps. the total - revenue function is r(x)=24x. (simplify your answer. do not include the $ symbol in your answer.) c) find the total profit p(x) from the production and sale of x dozen caps. the total - profit function is p(x)=18x - 25038. (simplify your answer. do not include the $ symbol in your answer.) d) find the profit or loss from the production and sale of 3000 dozen caps and 1000 dozen caps. the production and sale of 3000 dozen caps will result in a profit of $28962. the production and sale of 1000 dozen caps will result in a loss of $7038. e) find the break - even point. the break - even point is (type an ordered pair using integers or decimals. do not include the $ symbol in your answer.)

Answer

Explanation:

Step1: Recall break - even formula

At break - even, $P(x)=0$, where $P(x)=R(x)-C(x)$. Given $R(x) = 24x$ and $P(x)=18x - 25038$, and $P(x)=R(x)-C(x)$, we can find $C(x)$. First, since $P(x)=R(x)-C(x)$, then $C(x)=R(x)-P(x)$. Substituting the given functions: $C(x)=24x-(18x - 25038)=6x + 25038$. At break - even, $R(x)=C(x)$, so $24x=6x + 25038$.

Step2: Solve for $x$

Subtract $6x$ from both sides of the equation $24x=6x + 25038$. We get $24x-6x=25038$, which simplifies to $18x=25038$. Then divide both sides by 18: $x=\frac{25038}{18}=1391$.

Step3: Find the break - even point

The break - even point is the pair $(x, R(x))$ (or $(x,C(x))$ since they are equal at break - even). When $x = 1391$, $R(x)=24x=24\times1391 = 33384$. The break - even point is $(1391,33384)$. But we are asked to give the value of $x$ for the break - even point in the format of the problem.

Answer:

1391