Optimising Functions

Find two positive real numbers whose sum is exactly $60$ and whose product is as large as possible.

In the figure, the segment of length $60$ is split into two parts, $x$ and $y$. The splitting point is drawn at an arbitrary position: it does not indicate the solution.

A young livestock farmer has $120$ metres of fencing to enclose a rectangular area for grazing. Taking advantage of the fact that one side of the enclosure will be an existing stone wall, he only needs to fence the other three sides. Calculate the dimensions of the enclosure so that the enclosed area is maximised.

An animal shelter has received a donation of $120$ metres of wire mesh to build two adjacent rectangular enclosures for dogs and cats. The enclosures must share a common side, as shown in the figure, to make the best use of the material. Determine the dimensions of the outer structure ($x$ and $y$) so that the total enclosed area is maximised.

A chemical company needs to manufacture closed cylindrical tanks with a capacity of $16 \pi$ cubic metres. The material used for the bases (top and bottom) costs $5$ € per square metre, while the material for the curved surface is lighter and costs $2$ € per square metre. Calculate the radius of the base and the height of the cylinder that minimise the total manufacturing cost of each tank.

A capsule for a new drug needs to be designed in the shape of a right circular cylinder extended at its ends by two hemispheres of the same radius as the cylinder. For chemical stability reasons, the total volume of the capsule must be exactly $36\pi \text{ mm}^3$. The material used for the hemispherical ends is 50% more expensive per unit surface area than the material used for the cylindrical part. Determine the dimensions of the capsule (radius $r$ and length of the cylindrical part $h$) that minimise the total manufacturing cost.

A streaming platform is analysing its pricing policy for a new premium plan. Currently, they have $10.000$ subscribers who pay $12$ € per month. Market research indicates that for every euro the subscription price is reduced, $1.000$ new users will join. Conversely, for every euro the price increases, $1.000$ users will be lost. Knowing that the fixed maintenance costs are $20.000$ € per month and that each subscriber generates an administrative cost of $2$ €, what monthly subscription price will maximise the platform's net profit?

A logistics company is building a warehouse with an area of $400 \text{ m}^2$. The warehouse must be rectangular and divided into two sectors by an internal wall parallel to one of its side walls. The cost of the outer walls is $120 \text{ €}$ per linear metre, while the internal wall, being less strong, costs $80 \text{ €}$ per linear metre. Find the dimensions of the warehouse that make the cost of the walls as low as possible.

In a strategy video game, a scout unit is located at point $A(0, 5)$ on a coordinate map. There is a path whose trajectory follows the shape of the parabola $y = x^2$. The programmers need to calculate the exact points on the path that are at the minimum distance from the unit to trigger a proximity event. Determine the coordinates of these points.

A surveillance drone must travel from a base located at point $A(0, 4)$ to a target at point $B(10, 0)$, where the coordinates are expressed in kilometers. The terrain is divided into two zones by the line $y = 2$. In the upper zone ($y > 2$), due to wind conditions, the drone can fly at a speed of $60 \text{ km/h}$. In the lower zone ($y \leq 2$), the drone can reach $100 \text{ km/h}$. State the function that models the total travel time and the equation that determines the point on the dividing line through which the drone must cross so that this time is minimal (you do not need to solve it).