Answer
The number of computers is $23$ , and the maximum daily profit is $\$16,900$.
Work Step by Step
On comparing the given profit function $f\left( x \right)=-{{x}^{2}}+46x-360$ with the standard form of the quadratic function $f\left( x \right)=a{{x}^{2}}+bx+c$ , it can be observed that:
$a=-1,b=46,c=-360$
Since $a<0$ , the parabola opens downwards and thereby the function has a maximum value.
Then, the coordinates of the vertex of the parabola will be the coordinates of the maxima of the parabola. Thus, the maximum value occurs at,
$x=-\frac{b}{2a}=-\frac{46}{2\left( -1 \right)}=\frac{46}{2}=23$
Thus, the maximum profit occurs when 23 computers are manufactured. Then, the maximum profit is given by:
$\begin{align}
& f\left( 23 \right)=-{{\left( 23 \right)}^{2}}+46\left( 23 \right)-360 \\
& \text{ }=-529+1058-360 \\
& \text{ }=169
\end{align}$
Therefore, 23 computers should be manufactured each day to maximize the daily profit, which is $\$16,900$.
