# Consider a dress manufacturing company, where the cost of producing x amount of suits is: C(x)=841+18x^0.5 At which rate does the averege cost change after 500 have been produced?

Question
Analysis
Consider a dress manufacturing company, where the cost of producing x amount of suits is:
$$\displaystyle{C}{\left({x}\right)}={841}+{18}{x}^{{0.5}}$$
At which rate does the averege cost change after 500 have been produced?

2021-02-02
Take the derivative of the equation to find the rate of change of the equation:
$$\displaystyle{C}'{\left({x}\right)}={\left(\frac{{1}}{{2}}\right)}\times{18}{x}^{{\frac{{1}}{{2}}-{1}}}$$
Simplify:
$$\displaystyle{C}'{\left({x}\right)}={9}{x}^{{-\frac{{1}}{{2}}}}$$
Substitute x=500:
$$\displaystyle{C}'{\left({500}\right)}={9}{\left[{500}\right)}^{{-\frac{{1}}{{2}}}}{]}$$
simplify:
$$\displaystyle{C}'{\left({500}\right)}\approx{0.4025}$$
=40.25%

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