question a home improvement store is analyzing the pricing of a line of lawnmowers. these are the cost and…

question a home improvement store is analyzing the pricing of a line of lawnmowers. these are the cost and revenue functions for a month of business, where x represents the selling price of a lawnmower: r(x)= -2.075x² + 768x c(x)= -134.625x + 8,4037.5 which selling prices result in a profit that is greater than or equal to zero? select all the correct answers. 120 135 285 300 335

question a home improvement store is analyzing the pricing of a line of lawnmowers. these are the cost and revenue functions for a month of business, where x represents the selling price of a lawnmower: r(x)= -2.075x² + 768x c(x)= -134.625x + 8,4037.5 which selling prices result in a profit that is greater than or equal to zero? select all the correct answers. 120 135 285 300 335

Answer

Explanation:

Step1: Define profit function

Profit $P(x)=R(x)-C(x)$. So $P(x)= - 2.075x^{2}+768x-(-134.625x + 8403.75)=-2.075x^{2}+902.625x - 8403.75$.

Step2: Set profit $\geq0$

We want $-2.075x^{2}+902.625x - 8403.75\geq0$. Multiply through by - 1 to get $2.075x^{2}-902.625x + 8403.75\leq0$.

Step3: Use quadratic formula

For $ax^{2}+bx + c = 0$ ($a = 2.075$, $b=-902.625$, $c = 8403.75$), the quadratic formula is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(-902.625)^{2}-4\times2.075\times8403.75$. $\Delta = 814735.5625-4\times2.075\times8403.75=814735.5625 - 69451.5=745284.0625$. Then $x=\frac{902.625\pm\sqrt{745284.0625}}{2\times2.075}=\frac{902.625\pm863.29}{4.15}$. We get $x_1=\frac{902.625 + 863.29}{4.15}=\frac{1765.915}{4.15}\approx425.52$ and $x_2=\frac{902.625-863.29}{4.15}=\frac{39.335}{4.15}\approx9.48$. The solution to $2.075x^{2}-902.625x + 8403.75\leq0$ is $9.48\leq x\leq425.52$.

Answer:

120, 135, 285, 300, 335