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.

We wish to construct an open box (without a lid) from a square sheet of cardboard of side length $30 \text{ cm}$. To do this, a square of side $x$ is cut out from each corner and the edges are folded upwards. Express the volume of the box as a function of $x$ and determine the value of $x$ for which the volume of the box is maximum.

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 operates a daily round-trip route between Dublin and Galway along the M2 motorway, with a total travel distance of $640\text{ km}$. The average cruising speed $v$ (in $\text{km/h}$) must remain constant throughout the journey and, due to traffic regulations for commercial vehicles and road flow conditions, must lie in the interval $[70, 110]$.

The fuel consumption of the van (in liters of diesel per $100\text{ km}$) depends on the speed according to the empirical function $C(v) = 5.5 + 0.00075\times v^2$. The price of diesel on this route is $1.45\text{ €/L}$. In addition, the company computes an operating and driver wage cost of $19\text{ €}$ for each hour of travel.

1. a) Model the total cost function of the trip, $K(v)$, by adding the cost of fuel consumed and the hourly labor cost as a function of the speed $v$.
2. b) Find the average speed that minimizes the total cost of the trip. Rigorously justify that this is a relative minimum by analyzing the sign of the second derivative $K''(v)$.
3. c) Evaluate and compare the total cost at the endpoints of the legal domain ($v = 70\text{ km/h}$ and $v = 110\text{ km/h}$) with the minimum found, and state the optimal total cost of the trip.

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 per linear metre, while the internal wall, being less strong, costs €80 per linear metre. Find the dimensions of the warehouse that make the cost of the walls as low as possible.

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).

EirGrid is planning the power evacuation line from an offshore substation $S$ belonging to an offshore wind farm off the coast of County Galway. Substation $S$ is located at sea, at a perpendicular distance of $6\text{ km}$ from the straight coastline at point $A$. The onshore receiving substation $B$ is located on the same coastline, at a distance of $15\text{ km}$ from $A$.

The electrical cable will run along the seabed from $S$ to a landing point $P$ located on the coast between $A$ and $B$ (at a distance $x$ from $A$, with $0 \le x \le 15$), and will continue underground from $P$ to $B$ along the coastline. The installation cost of the submarine cable is $1.5\text{ million euros/km}$ and that of the overland cable is $0.9\text{ million euros/km}$.

1. a) Formulate the total cost function $C(x)$ in millions of euros and define its physical domain $[0, 15]$.
2. b) Calculate the position of $P$ (value of $x$) that minimizes the total cost and justify that it is the absolute minimum by comparing the obtained value with the endpoints of the closed interval.
3. c) If $\theta$ represents the angle formed by the submarine route $SP$ with the perpendicular to the coast, prove that at the optimal point Snell's Law is exactly satisfied: $\frac{\sin \theta}{c_{\text{land}}} = \frac{1}{c_{\text{sea}}}$, where $c_{\text{sea}} = 1.5$ and $c_{\text{land}} = 0.9$.
4. d) Transfer and threshold problem: Suppose that the submarine cost is reduced to a generic value $c$ (keeping the overland cost at $0.9\text{ million/km}$). Determine above/below which critical value of $c$ an intermediate landing point $P \in (0, 15)$ will cease to exist and it will be cheaper to run the cable in a direct straight line through the sea from $S$ to $B$. Justify your reasoning by studying the sign of $C'(x)$ at the upper endpoint of the domain.

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.

You want to design an advertising panel in the shape of an isosceles triangle inscribed in a semicircle of radius $R = 6$ meters. One of its vertices is at the center $O$ of the semicircle, and the other two vertices lie on the arc, joined by a chord $b$ (the base) parallel to the diameter. Call $h$ the height of the triangle, that is, the distance from $O$ to the base, with $0 < h < 6$.

Determine the dimensions of the triangle (base and height) so that the area of the advertising surface is maximum.

A rectangular solar panel is to be installed on a wall shaped like an isosceles triangle, with a base of $6 \text{ m}$ and a height of $4 \text{ m}$. The panel must be placed so that its base lies on the base of the triangle and its two top vertices touch the equal sides of the triangle. However, due to the structure of the wall, the panel has an efficiency that varies with its height, such that the total energy generated is given by the function $E = A \times (5 - h)$, where $A$ is the area of the panel and $h$ is its height. Find the dimensions of the panel that maximise the energy generated.