A shop sells 500 smartphones a week for $\$450$ each. A market survey shows that each decrease of $\$5$ on the price will result in the sale of an additional 10 smartphones per week. What price of the smartphone would result in maximum revenue, in dollars?
Suppose the price of the cellphone is reduced to $450 - 5x$ dollars; then it will sell $500 + 10x$ units, so the revenue will be
\begin{align*}
(450 - 5x)(500 + 10x) &= 5(90 - x) 10(50 + x) \\
&= 50 (90 - x)(50 + x) \\
&= 50 (-x^2  + 40x + 4500),
\end{align*}in dollars.

Completing the square on $-x^2 + 40x + 4500,$ we get
\begin{align*}
50 (-x^2 + 40x + 4500) &= 50 (-(x - 20)^2 + 400 + 4500) \\
&= 50 (-(x - 20)^2 + 4900) \\
&= -50 (x - 20)^2 + 245000.
\end{align*}This is maximized when $x = 20,$ so the optimal price of the smartphone is $450 - 5(20) = \boxed{350}$ dollars.