Question

A company's profit C (in thousands of dollars) can be modeled by the polynomial function C = - 5x ^ 3 + 6x ^ 2 + 15x , where x is the number of items produced in thousands. The profit is $14,000 for producing 2000 items. What other number of items would produce about the same profit?

Answer 1

Answer:

849 items.

Step-by-step explanation:

Given that the profit C (in thousands of dollars) for x thousands of items related as

$C = - 5x ^ 3 + 6x ^ 2 + 15x \\\\\Rightarrow -5x^3+6x^2+15x -C=0\cdots(i)$

As the profit is $14,000 for producing 2000 items, so

C= 14 thousand dollars and

x= 2 thousand items.

Putting C= 14 in the equation ( we have),

$-5x^3+6x^2+15x -14=0\cdots(ii)$

Now, x=2 is one of the solutions to the equation (ii), so (x-2) is a factor of the equation (ii), we have

$(x-2)(-5x^2-4x+7)=0 \\\\\Rightarrow x-2=2 \; or \; -5x^2-4x+7=0$

We have the given solution for x-2=0, so sloving -5x^2-4x+7=0 for other solutions.

$-5x^2-4x+7=0 \\\\\Rightarrow x= \frac {-(-4)\pm \sqrt {(-4)^2-4\times (-5)7}}{2\times (-5)} \\\\\Rightarrow x= \frac {4\pm \sqrt {156}}{2\times (-5)} \\\\\Rightarrow x= \frac {4\pm 12.49}{2\times (-5)} \\\\\Rightarrow x = \frac {4+ 12.49}{2\times (-5)}, \frac {4- 12.49}{2\times (-5)} \\\\\Rightarrow x = -1.649, 0.849$

As the number of items cant be negative, so x= 0.849 thousand is the other number of items.

Hence, the other number of items for the same profit is 849 items.