Solve the problem.
-The total profit function $P ( x )$ for a company producing $x$ thousand units is given by $P ( x ) = - 2 x ^ { 2 } + 30 x - 100$. 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 between 5 thousand units and 10 thousand units
B) $x$ is greater than 5 thousand units
C) $x$ is less than 10 thousand units
D) $x$ is less than 5 thousand units or greater than 10 thousand units

Question

Solve the problem.
-The total profit function $P ( x )$ for a company producing $x$ thousand units is given by $P ( x ) = - 2 x ^ { 2 } + 30 x - 100$. 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 between 5 thousand units and 10 thousand units
B) $x$ is greater than 5 thousand units
C) $x$ is less than 10 thousand units
D) $x$ is less than 5 thousand units or greater than 10 thousand units

Quizplus · Accepted Answer

To find the values of $x$ for which $P(x) > 0$, we solve the inequality $-2x^2 + 30x - 100 > 0$. Factoring or using the quadratic formula, we find the roots of the corresponding equation $-2x^2 + 30x - 100 = 0$ to be $x = 5$ and $x = 10$. Since the coefficient of $x^2$ is negative, the parabola opens downwards, meaning $P(x) > 0$ between the roots. Thus, the company makes a profit when $x$ is between 5 thousand units and 10 thousand units.

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