Question

The total cost (in dollars) for a company to manufacture and sell x items per week is C=40x+180, whereas the revenue brought in by selling all x items is R=68x−0.4x2. How many items must be sold to obtain a weekly profit of $300?

Answer 1

Answer:

The company needs to sell either 30 or 40 items.

Step-by-step explanation:

We are given that the cost for selling x items given by the function:

$C(x)=40x+180$

And the revenue for selling x items is given by:

$R(x)=68x-0.4x^2$

The profit function is the cost function subtracted from the revenue function:

$P(x)=R(x)-C(x)$

Substitute and simplify:

$\displaystyle \begin{aligned} P(x)&=(68x-0.4x^2)-(40x+180)\\&=68x-0.4x^2-40x-180\\&=-0.4x^2+28x-180\end{aligned}$

To find how many items must be sold in order to obtain a weekly profit of $300, we can let P equal 300 and solve for x. So:

$300=-0.4x^2+28x-180$

Solve for x. Subtract 300 from both sides:

$-0.4x^2+28x-480=0$

We can divide both sides by -0.4:

$x^2-70x+1200=0$

Factor:

$(x-40)(x-30)=0$

Zero Product Property:

$x-40=0\text{ or } x-30=0$

Solve for each case:

$x=40\text{ or } x=30$

So, in order to obtain a weekly profit of $300, the company need to sell either 30 or 40 items.