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. it costs adele $2 to make each container of trail mix. so, she will earn x - 2 dollars in profit per container. which equation can adele use to determine the price she can charge per container to earn $1,500 in profit in her first month? 1,500=(x - 2)(-20x + 413) 1,500x=(x - 2)(-20x + 413) to the nearest dollar, what is the lowest price adele can charge for each container to earn $1,500 in profit in her first month? $

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. it costs adele $2 to make each container of trail mix. so, she will earn x - 2 dollars in profit per container. which equation can adele use to determine the price she can charge per container to earn $1,500 in profit in her first month? 1,500=(x - 2)(-20x + 413) 1,500x=(x - 2)(-20x + 413) to the nearest dollar, what is the lowest price adele can charge for each container to earn $1,500 in profit in her first month? $

Answer

Explanation:

Step1: Recall profit - formula

Profit = Profit per unit×Number of units sold.

Step2: Identify profit - per - unit and number of units

Profit per container is $x - 2$ dollars and the number of containers sold is $-20x + 413$. The target profit is $1500$ dollars. So, $1500=(x - 2)(-20x + 413)$.

Step3: Expand the right - hand side

Expand $(x - 2)(-20x + 413)$ using the 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 equation becomes $1500=-20x^{2}+453x - 826$, or $20x^{2}-453x + 2326 = 0$.

Step4: Use the quadratic formula

The quadratic formula for $ax^{2}+bx + c = 0$ is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a = 20$, $b=-453$, and $c = 2326$. First, calculate the discriminant $\Delta=b^{2}-4ac=(-453)^{2}-4\times20\times2326=205209-186080 = 19129$. Then $x=\frac{453\pm\sqrt{19129}}{40}=\frac{453\pm138.31}{40}$. We have two solutions: $x_1=\frac{453 + 138.31}{40}=\frac{591.31}{40}=14.78275$ and $x_2=\frac{453-138.31}{40}=\frac{314.69}{40}=7.86725$.

Answer:

$8$