Find all numbers that satisfy the following.
-The total profit function $\mathrm { P } ( \mathrm { x } )$ for a company producing $x$ thousand units is given by $P ( x ) = - 3 x ^ { 2 } + 45 x - 162$. Find the values of $x$ for which the company makes a profit. [Hint: The company makes a profit when $\mathrm { P } ( \mathrm { x } ) 0$.]
A) $x$ is greater than 6 thousand units
B) $x$ is less than 9 thousand units
C) $x$ is between 6 thousand units and 9 thousand units
D) $x$ is less than 6 thousand units or greater than 9 thousand units

Question

Find all numbers that satisfy the following.
-The total profit function $\mathrm { P } ( \mathrm { x } )$ for a company producing $x$ thousand units is given by $P ( x ) = - 3 x ^ { 2 } + 45 x - 162$. Find the values of $x$ for which the company makes a profit. [Hint: The company makes a profit when $\mathrm { P } ( \mathrm { x } ) > 0$.]
A) $x$ is greater than 6 thousand units
B) $x$ is less than 9 thousand units
C) $x$ is between 6 thousand units and 9 thousand units
D) $x$ is less than 6 thousand units or greater than 9 thousand units

Quizplus · Accepted Answer

To find the values of $x$ for which the company makes a profit, we need to solve the inequality $P(x) > 0$, which is $-3x^2 + 45x - 162 > 0$. Factoring or using the quadratic formula, we find the roots of the equation $P(x) = 0$ to be $x = 6$ and $x = 9$. Since $P(x)$ is a downward-opening parabola (because the coefficient of $x^2$ is negative), the function $P(x)$ is positive between these two roots. Therefore, the company makes a profit when $x$ is between 6 thousand units and 9 thousand units.

Find All Numbers That Satisfy the Following P(x)\mathrm { P } ( \mathrm { x } )P(x)

Find All Numbers That Satisfy the Following $\mathrm { P } ( \mathrm { x } )$