adele recently decided to start selling her homemade trail mix online. she did some market research to…

adele recently decided to start selling her homemade trail mix online. she did some market research to predict how the price she charges per container will affect the number of containers she sells. she finds that if she charges x dollars per container, she will sell -20x + 413 containers of trail mix in her first month.\nit costs adele $2 to make each container of trail mix. so, she will earn x - 2 dollars in profit per container.\nwhich equation can adele use to determine the price she can charge per container to earn $1,500 in profit in her first month?\n1,500x=(x - 2)(-20x + 413)\n1,500=(x - 2)(-20x + 413)\nto the nearest dollar, what is the lowest price adele can charge for each container to earn $1,500 in profit in her first month?\n$

adele recently decided to start selling her homemade trail mix online. she did some market research to predict how the price she charges per container will affect the number of containers she sells. she finds that if she charges x dollars per container, she will sell -20x + 413 containers of trail mix in her first month.\nit costs adele $2 to make each container of trail mix. so, she will earn x - 2 dollars in profit per container.\nwhich equation can adele use to determine the price she can charge per container to earn $1,500 in profit in her first month?\n1,500x=(x - 2)(-20x + 413)\n1,500=(x - 2)(-20x + 413)\nto the nearest dollar, what is the lowest price adele can charge for each container to earn $1,500 in profit in her first month?\n$

Answer

Explanation:

Step1: Identify profit - formula

Profit = Profit per unit×Number of units. Profit per unit is $x - 2$, number of units is $-20x + 413$, and total profit is $1500$. So the equation is $1500=(x - 2)(-20x + 413)$.

Step2: Expand the right - hand side

Expand $(x - 2)(-20x + 413)$ using FOIL method: [ \begin{align*} (x - 2)(-20x + 413)&=x\times(-20x)+x\times413-2\times(-20x)-2\times413\ &=-20x^{2}+413x + 40x-826\ &=-20x^{2}+453x-826 \end{align*} ] So the quadratic equation is $-20x^{2}+453x - 826=1500$, or $-20x^{2}+453x-2326 = 0$. For a quadratic equation $ax^{2}+bx + c = 0$ ($a=-20$, $b = 453$, $c=-2326$), the quadratic formula is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. [ \begin{align*} x&=\frac{-453\pm\sqrt{453^{2}-4\times(-20)\times(-2326)}}{2\times(-20)}\ &=\frac{-453\pm\sqrt{205209-186080}}{-40}\ &=\frac{-453\pm\sqrt{19129}}{-40}\ &=\frac{-453\pm138.31}{-40} \end{align*} ] We have two solutions: $x_1=\frac{-453 + 138.31}{-40}=\frac{-314.69}{-40}=7.86725$ $x_2=\frac{-453-138.31}{-40}=\frac{-591.31}{-40}=14.78275$

Answer:

The correct equation is $1,500=(x - 2)(-20x + 413)$. The lowest price to the nearest dollar is $$8$.