Question

Suppose that in a monopoly market the total cost per week of producing a high tech product is given by C(x) =3600+100x+2x^2.

Suppose further that the weekly demand function for this product is p=500-2x.

Find the number of units that will give the break even points for the product.

Find the number of units that will give maximum profit.

Answer 1

Answer:

10 units;

50 units.

Explanation:

The revenue function is given by the price function multiplied by the number of units sold (x).

$R(x) = x*p(x)=500x-2x^2$

The break even point occurs when Revenue equals costs:

$C(x) =R(x)\\3600+100x+2x^2=500x-2x^2\\4x^2-400x+3600 = 0\\x^2-100x+900=0\\x=\frac{100\pm \sqrt{100^2-(4*900)}}{2}\\x_1=10\\x_2=90$

Therefore, the smallest number of units required for the company to break even is 10 units.

Maximum profit will be achieved at that number of units for which the derivate of the profit function is zero:

$P(x)=R(x) - C(x) \\P(x)=3600+100x+2x^2-(500x-2x^2)\\\frac{dP(x)}{dx}=0=8x-400\\x=50$

The number of units that will give maximum profit is 50.