Question

The demand for a product, in dollars, is p = 2000-0.2x-0.01x² find the consumer surplus when the sales level is 250

Answer 1

Since the number of products sold is $x=250,$ the coresponding price is

$$P=2000-0.2(250)-0.01(250)^2=1325$$

The consumer surplus is as follows

$$\displaystyle\int_{0}^{250}(p(x)-P)dx$$

$$=\displaystyle\int_{0}^{250}(2000-0.2x-0.01x^2-1325)dx$$

$$=[675x-0.1x^2-\dfrac{0.01x^3}{3}]\begin{matrix} 250 \\ 0 \end{matrix}$$

$$=675(250)-0.1(250)^2-\dfrac{0.01(250)^3}{3}$$

$$=\$110416.67$$