To solve this problem, we need to accurately interpret the function [tex]\(P(x) = -0.0005(x^2 + 30)(x - 20)(x - 70)\)[/tex], which models the weekly profit earned by the clothing company when it makes and sells [tex]\(x\)[/tex] jackets.
### When the Profit is Exactly [tex]$0:
The company’s profit will be exactly \$[/tex]0 when [tex]\(P(x) = 0\)[/tex]. To find the values of [tex]\(x\)[/tex] where [tex]\(P(x) = 0\)[/tex], we need to determine the roots of the equation [tex]\( -0.0005(x^2 + 30)(x - 20)(x - 70) = 0 \)[/tex].
The expression will be zero if any of its factors are zero:
1. [tex]\(x^2 + 30 = 0\)[/tex]
2. [tex]\(x - 20 = 0\)[/tex]
3. [tex]\(x - 70 = 0\)[/tex]
- [tex]\(x^2 + 30 = 0\)[/tex]: This equation has no real solutions because [tex]\(x^2\)[/tex] is always non-negative and thus cannot equal [tex]\(-30\)[/tex] since [tex]\(-30\)[/tex] is negative.
- [tex]\(x - 20 = 0\)[/tex]: Solving for [tex]\(x\)[/tex] gives [tex]\(x = 20\)[/tex].
- [tex]\(x - 70 = 0\)[/tex]: Solving for [tex]\(x\)[/tex] gives [tex]\(x = 70\)[/tex].
Therefore, the company’s profit will be exactly \[tex]$0 if it makes and sells 20 jackets or 70 jackets.
### When the Profit is Positive:
To find the range of \(x\) where the company makes a profit, we need to consider the behavior of the polynomial function. By examining the nature of the cubic polynomial \(P(x)\), we understand that the function will be positive between its real roots and negative outside these roots due to the negative leading coefficient.
Thus, the company will make a profit when the function \(P(x) > 0\). This occurs when \(20 < x < 70\). Outside this interval, the profit will be negative.
- The company makes a profit if it makes and sells jackets between 20 and 70 (excluding 20 and 70).
- The company does not make a profit if it makes and sells either less than 20 jackets or more than 70 jackets.
### Summary:
1. The company's profit will be exactly $[/tex]0 if it makes and sells [tex]\(20\)[/tex] jackets or [tex]\(70\)[/tex] jackets.
2. The company will make a profit if it makes and sells between [tex]\(20\)[/tex] and [tex]\(70\)[/tex] jackets (not including [tex]\(20\)[/tex] and [tex]\(70\)[/tex]).
3. The company will not make a profit if it makes and sells less than [tex]\(20\)[/tex] jackets or more than [tex]\(70\)[/tex] jackets.
