question 19\nevaluating and solving quadratic functions\na companys revenue earned from selling x items is…

question 19\nevaluating and solving quadratic functions\na companys revenue earned from selling x items is given by the function $r(x)=570x$, and their cost is given by the function $c(x)=1430 + 2.1x^{2}$. use this function to answer the following questions.\nwrite a function, p(x), that represents the companys profit from selling x items.\n$p(x)=$\nidentify the vertical intercept of p(x). write it as an ordered pair and interpret its meaning in a complete sentence.\nvertical intercept:\nif the company sells\nhow many items must be sold in order to maximize the profit?\nto maximize profits,\nwhat is the maximum profit that can be earned? round to the nearest cent.\nthe maximum profit that can be earned is $$\nwhat is the minimum number of items that must be sold in order to break even? round to the nearest whole number.\nthe company must sell a minimum of\nquestion help: video post to forum

question 19\nevaluating and solving quadratic functions\na companys revenue earned from selling x items is given by the function $r(x)=570x$, and their cost is given by the function $c(x)=1430 + 2.1x^{2}$. use this function to answer the following questions.\nwrite a function, p(x), that represents the companys profit from selling x items.\n$p(x)=$\nidentify the vertical intercept of p(x). write it as an ordered pair and interpret its meaning in a complete sentence.\nvertical intercept:\nif the company sells\nhow many items must be sold in order to maximize the profit?\nto maximize profits,\nwhat is the maximum profit that can be earned? round to the nearest cent.\nthe maximum profit that can be earned is $$\nwhat is the minimum number of items that must be sold in order to break even? round to the nearest whole number.\nthe company must sell a minimum of\nquestion help: video post to forum

Answer

Explanation:

Step1: Recall profit formula

Profit $P(x)=R(x)-C(x)$. Given $R(x) = 570x$ and $C(x)=1430 + 2.1x^{2}$, then $P(x)=570x-(1430 + 2.1x^{2})=- 2.1x^{2}+570x - 1430$.

Step2: Find vertical - intercept

The vertical - intercept of a function $y = P(x)$ is found by setting $x = 0$. So, $P(0)=-2.1(0)^{2}+570(0)-1430=-1430$. The ordered pair is $(0, - 1430)$. This means that when the company sells 0 items, they have a loss of $1430.

Step3: Find x for maximum of quadratic function

For a quadratic function $y = ax^{2}+bx + c$ ($a=-2.1$, $b = 570$, $c=-1430$), the x - value of the vertex (where the function is maximized since $a<0$) is given by $x=-\frac{b}{2a}$. So, $x=-\frac{570}{2\times(-2.1)}=\frac{570}{4.2}\approx136$.

Step4: Find maximum profit

Substitute $x\approx136$ into $P(x)=-2.1x^{2}+570x - 1430$. $P(136)=-2.1(136)^{2}+570(136)-1430=-2.1\times18496+77520 - 1430=-38841.6+77520 - 1430=37248.4$.

Step5: Find break - even points

Set $P(x)=0$, so $-2.1x^{2}+570x - 1430 = 0$. Using the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$, where $a=-2.1$, $b = 570$, $c=-1430$. First, calculate the discriminant $\Delta=b^{2}-4ac=(570)^{2}-4\times(-2.1)\times(-1430)=324900-12012 = 312888$. Then $x=\frac{-570\pm\sqrt{312888}}{-4.2}$. $x_1=\frac{-570+\sqrt{312888}}{-4.2}\approx\frac{-570 + 559.36}{-4.2}\approx2.53$ and $x_2=\frac{-570-\sqrt{312888}}{-4.2}\approx\frac{-570 - 559.36}{-4.2}\approx278.42$. The minimum number of items to break even is 3 (rounding up the non - zero root).

Answer:

$P(x)=-2.1x^{2}+570x - 1430$ Vertical Intercept: $(0, - 1430)$. When the company sells 0 items, they have a loss of $1430. To maximize profits, 136 items must be sold. The maximum profit that can be earned is $$37248.40$. The company must sell a minimum of 3 items to break even.