Answer:
(140, 30,072) and (380, 19608).
Step-by-step explanation:
We are given that the revenue function for selling the kitchen table given its price is:
[tex]R(x)=-0.68x^2+310x[/tex]
And the cost function is:
[tex]C(x)=-43.6x+36176[/tex]
At the breakeven point, the revenue equals the cost. Therefore:
[tex]R(x)=C(x)[/tex]
Substitute:
[tex]-0.68x^2+310x=-43.6x+36176[/tex]
We can solve for x.
Adding 0.68x² to both sides and subtracting 310x from both sides yields:
[tex]0.68x^2-353.6x+36176=0[/tex]
We can use the quadratic formula:
[tex]\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
In this case, a = 0.68, b = -353.6, and c = 36176. Substitute:
[tex]\displaystyle x=\frac{-(-353.6)\pm\sqrt{(-353.6)^2-4(0.68)(36176)}}{2(0.68)}[/tex]
Simplify:
[tex]\displaystyle x=\frac{353.6\pm\sqrt{26684.24}}{1.36}[/tex]
So, our two solutions are:
[tex]\displaystyle x=\frac{353.6+\sqrt{26684.24}}{1.36}\text{ or } x=\frac{353.6-\sqrt{26684.24}}{1.36}[/tex]
Use a calculator:
[tex]x\approx 380\text{ or } x\approx 140[/tex]
Our answers are:
(140, 30,072) and (380, 19608).
