A small bakery makes a profit $P(c)$ in dollars from selling $c$ cakes per day according to the formula $P(c) = c(40 - c) - 80$. Considering that $P$ is a continuous function, how many cakes should the bakery sell per day to maximize its profit?

Given the objective function $Z=4x+3y$, with the constraint that $xy=72$, for $x>0$ and $y>0$. Determine the absolute minimum value of $Z$.

Determine the minimum sum $x+4y$ for positive numbers $x$ and $y$ satisfying the equation $xy=18$.

In a park, a zip line is to be installed from the top of a tower to the ground and then up to the top of a hill for an adventure activity. The tower is $60$ meters high, the hill is $40$ meters high, and the horizontal distance between them is $100$ meters. To minimize the length of the cable used, what should be the angle between the cable and the horizontal at the point where the cable touches the ground? Report your final answer in units of degrees rounded to two decimal places.

Find the production level that will minimize the average cost for a factory where the cost of producing $x$ units of the product is given by $c\left(x\right)=x^3-40x^2+16000x$.

In an optical experiment, a beam of light travels from a source located at point $E$ to a target at point $F$. The beam encounters a flat mirror at point $O$. To ensure the beam follows the quickest route from $E$ to $F$ after reflecting off the mirror, what must be true about the angles formed with the normal at the reflection point?