Answer
a) $P(x)=12.5x^{4}+10x^{3}-40$
b)-$\$240$
Work Step by Step
a)
The profit function is:
$$P(x)=\int P'(x)dx$$
$$P(x)=\int x(50x^{2}+30x)dx$$
$$P(x)=\int (50x^{3}+30x^{2})dx$$
$$P(x)=\frac{50}{4}x^{4}+\frac{30}{3}x^{3}+C$$
$$P(x)=12.5x^{4}+10x^{3}+ C$$
We have for $x=0$, the value of $P$ is $-40$, so:
$$P(0)=12.5\cdot 0^{4}+10\cdot 0^{3}+ C$$
$$-40=12.5\cdot 0^{4}+10\cdot 0^{3}+ C$$
$$-40= C$$
Therefore, the profit function is:
$$P(x)=12.5x^{4}+10x^{3}-40$$
b)
$$P(2)=12.5(2)^{4}+10(2)^{3}-40=$240$$
