a company manufactures two products. market research and available resources require the following…

a company manufactures two products. market research and available resources require the following constraints:\n- the number of units of product a manufactured, $x$, is at most 500 units more than twice the number of units of product b, $y$.\n- the square of the companys profit is equal to the sum of 35 times the number of product a units sold and 50 times the number of product b units sold.\nif the company expects weekly profits to exceed $22,500, which pair of inequalities represents these constraints?\n$xleq 2y + 500$\n$sqrt{35x + 50y}>150$\n$xgeq 2y + 500$\n$sqrt{35x + 50y}>22,500$\n$xleq 2y + 500$\n$sqrt{35x + 50y}>22,500$\n$xleq 2y + 500$\n$35x + 50y>22,500$

a company manufactures two products. market research and available resources require the following constraints:\n- the number of units of product a manufactured, $x$, is at most 500 units more than twice the number of units of product b, $y$.\n- the square of the companys profit is equal to the sum of 35 times the number of product a units sold and 50 times the number of product b units sold.\nif the company expects weekly profits to exceed $22,500, which pair of inequalities represents these constraints?\n$xleq 2y + 500$\n$sqrt{35x + 50y}>150$\n$xgeq 2y + 500$\n$sqrt{35x + 50y}>22,500$\n$xleq 2y + 500$\n$sqrt{35x + 50y}>22,500$\n$xleq 2y + 500$\n$35x + 50y>22,500$

Answer

Explanation:

Step1: Translate the first constraint

The number of units of product A ($x$) is at most 500 units more than twice the number of units of product B ($y$). "At most" means $\leq$, so the inequality is $x\leq 2y + 500$.

Step2: Translate the profit - related constraint

Let $P$ be the profit. We know that $P^{2}=35x + 50y$. Since the company expects weekly profits to exceed $$22500$, then $P>22500$. Substituting $P = \sqrt{35x + 50y}$ (since $P^{2}=35x + 50y$ and $P>0$ for profit), we get $\sqrt{35x+50y}>22500$.

Answer:

$x\leq 2y + 500$, $\sqrt{35x + 50y}>22500$