Exercises: Function optimisation

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.

A small weather probe is launched vertically upwards. The height $h$ (in metres) reached by the probe as a function of time $t$ (in seconds) is given by the expression $h(t) = -5t^2 + 40t + 1.5$.

At what time will the probe reach its maximum height? What will this height be?

A technology company has determined that the monthly profit $B$ (in thousands of euros) from the sale of a new model of wireless headphones depends on the number of units manufactured $x$ (in hundreds) according to the function $B(x) = -2x^2 + 32x - 40$.

Find the number of units that must be manufactured each month to maximise profit, and calculate the value of this maximum profit.

A graphic design studio wants to create a rectangular greeting card that contains exactly $32 \text{ cm}^2$ of printed surface area. If margins are not considered, what should the dimensions of the card be so that the perimetre is minimised, thus saving on paper cutting?

The number of "likes" on a famous influencer's post during the first $10$ hours follows the function $L(t) = -t^3 + 12t^2 + 45t$, where $t$ is the elapsed time in hours ($0 \le t \le 10$). Determine at what moment the post reaches its peak success (maximum number of likes) and how many likes it gets at that moment.

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.

Divide the number $12$ into two positive parts such that the sum of their squares is minimum.

Cian needs to solve the following optimization problem: "An open-top box without a lid is to be constructed from a square piece of cardboard with a side length of $30\text{ cm}$, by cutting four equal squares of side length $x$ out of each corner and folding the flaps up. Find the value of $x$ that maximizes the volume of the box."

Cian writes on his exam:

1. Volume: $V(x) = x(30 - 2x)^2 = 4x^3 - 120x^2 + 900x$.

2. Derivative: $V'(x) = 12x^2 - 240x + 900 = 0 \implies x^2 - 20x + 75 = 0$, whose solutions are $x = 15$ and $x = 5$.

3. Cian's conclusion: "The maximum volume is obtained for $x = 15\text{ cm}$ because $15 > 5$".

1. a) Identify and explain the two conceptual mistakes made by Cian: one concerning the physical domain of the variable $x$ and another concerning the criterion for classifying maxima and minima using the derivative.
2. b) Correct the exercise following the five steps: define the admissible physical domain, properly classify the critical points using the sign of $V''(x)$ or $V'(x)$, and calculate the dimensions of the box and its maximum volume.

A farmer from the Vega Baja region has a straight stone wall of length $30\text{ m}$ and has $50\text{ m}$ of wire mesh available to enclose a rectangular vegetable garden. He wants to attach the garden to the existing wall, so that the side resting against the wall requires no mesh, but its length $y$ cannot exceed the $30\text{ m}$ of available wall. The other three sides will be enclosed using the $50\text{ m}$ of mesh.

1. a) Estimate: Before performing any analytical calculation, answer intuitively: do you think it would be better to use the entire $30\text{ m}$ of the wall to "save" as much mesh as possible, or would it be preferable to leave part of the wall free to achieve a more square-like shape? Provide a reasoned estimate of the maximum area you expect to obtain.
2. b) Calculate: Express the area of the enclosure $A(x)$ in terms of the length $x$ of the two sides perpendicular to the wall. Determine the domain of definition of the function taking into account the physical restriction of the wall ($y = 50 - 2x \le 30$). Calculate the value of $x$ that makes the first derivative zero and classify it.
3. c) Check and reflect: Find the definitive optimal dimensions and the maximum area. Compare the result with your initial estimate and explain why the intuition of always using the entire available wall turns out to be misleading in this case.

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 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 graphic design studio wants to create a promotional packaging box from a square sheet of card with a side length of $24$ cm. To do this, a square of side $x$ is cut out from each corner, and the flaps are folded upwards to form an open box. Find the value of $x$ so that the capacity of the box is as large 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 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.

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 page of an art book must have a total area of $600$ $cm^2$. The top and bottom margins must be $3$ cm each, and the side margins $2$ cm each. Determine the dimensions of the page (width and height) so that the area of the printed region (where the text and images go) is maximised.

In a metal workshop, a steel beam with a rectangular cross-section is to be cut from a cylindrical log with a diameter of $40$ cm. The mechanical strength $R$ of this beam is proportional to the product of the width $x$ and the square of the height $y$, that is, $R = k \times x \times y^2$ (where $k$ is a material strength constant). Find the dimensions $x$ and $y$ that the beam must have so that its strength is maximised.

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.

The goal is to determine the point $P(x, y)$ on the parabola $y = x^2$ that is at the shortest possible distance from the external point $A(0, 2)$. To solve it, three possible methods are suggested:

• Method 1: Set up the Euclidean distance $d(x) = \sqrt{(x-0)^2 + (x^2-2)^2}$ and differentiate the square root directly using the chain rule.

• Method 2: Minimize the auxiliary function $f(x) = [d(x)]^2 = x^2 + (x^2-2)^2 = x^4 - 3x^2 + 4$.

• Method 3: Find the point $P(x, x^2)$ by requiring that the vector $\vec{AP} = (x, x^2 - 2)$ be perpendicular to the direction vector of the tangent line to the parabola at $P$, that is, $\vec{AP} \cdot (1, 2x) = 0$.

1. a) Formally justify why Method 2 guarantees exactly the same critical points and extrema as Method 1, explaining why it is algebraically preferable.
2. b) Apply Method 2 to find the candidates for a minimum, classify them using the sign of $f''(x)$, and rule out those that do not correspond to the minimum distance.
3. c) Solve using Method 3 and verify that the equation obtained is identical to that of Method 2. Which of the three methods would you choose in a university entrance exam for speed and lower risk of calculation error? Give the coordinates of the closest points and the minimum distance.

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 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).

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 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.

Consider the function $f(x) = \frac{1}{x^2}$ for $x > 0$. A tangent line to the graph of this function is drawn at a generic point $P(a, f(a))$. This tangent line forms, together with the coordinate axes in the first quadrant, a right triangle. Determine whether there is any value of $a$ for which the area of that triangle is minimal and, if so, calculate the value of that area.

A tin manufacturer wants to create a new cylindrical container with a volume of $500 \text{ cm}^3$. The material for the lid and base costs $0.02 \text{ €/cm}^2$, while the material for the side costs $0.01 \text{ €/cm}^2$. Furthermore, to reinforce the structure, a metal strip must be placed around the joint between the lid and the side, and another around the base, at a cost of $0.05 \text{ €/cm}$ for the strip. Calculate the radius and height of the tin that minimise the total production cost.

Two points $A$ and $B$ are located on the same side of a line $r$. The point $A$ is at a distance of $3 \text{ km}$ from the line, and the point $B$ is at $6 \text{ km}$. The distance between the projections of $A$ and $B$ onto the line $r$ is $12 \text{ km}$. We wish to find a point $C$ on the line $r$ such that the sum of the squares of the distances from $C$ to $A$ and from $C$ to $B$ is minimised. Prove that this point coincides with the midpoint of the projection of $AB$ onto the line.

We have a piece of wire of length $20 \text{ metres}$. We want to cut it into two pieces (not necessarily equal) to form a square with one piece and an equilateral triangle with the other. How should we make the cut so that the sum of the areas of both shapes is maximised? And so that it is minimised? Justify your answer by analysing the endpoints of the interval of definition of the variable.

An agricultural cooperative commissions the design of a vertical grain silo consisting of a right circular cylinder of radius $r$ and height $h$, closed at its bottom base by a flat circular bottom of radius $r$, and topped at its upper part by a hemispherical dome of radius $r$. Due to strength requirements, the manufacturing costs per square meter are:

• Flat bottom base: $20\text{ \euro{/m}}^2$.

• Lateral cylindrical wall: $30\text{ \euro{/m}}^2$.

• Hemispherical dome: $k\text{ \euro{/m}}^2$ (where $k$ is a positive unit cost set by the bending workshop).

The engineering team establishes as a design criterion that, regardless of the fixed grain volume $V_0 > 0$ that the silo must hold, the optimal dimensions $(r, h)$ that minimize the total cost must necessarily satisfy the aesthetic and structural relation $h = 2r$.

1. a) Write the total cost function $C(r)$ solely in terms of the radius $r$ and the constants $k$ and $V_0$, using the total volume constraint of the silo ($V_0 = \pi r^2 h + \frac{2}{3}\pi r^3$).
2. b) Differentiate the function $C(r)$ with respect to $r$ and set the critical point condition $C'(r) = 0$.
3. c) Determine what exact value the price per square meter $k$ of the hemisphere must have so that the optimal condition coincides with $h = 2r$. Analytically prove that this value of $k$ does not depend on the chosen volume $V_0$.
4. d) Check using the sign of $C''(r)$ that the solution found indeed corresponds to an absolute minimum of the cost.

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.

Unit challenge. How does our country manage its power grid? You are going to advise an energy cooperative by analysing real Irish electricity demand.

1. Data retrieval and modelling: Go to the EirGrid Smart Grid Dashboard and download the hourly demand data ($24\text{ h}$) from a recent day. In GeoGebra or a spreadsheet, plot the scatter plot and perform a polynomial fit to obtain $D(t)$ (with $0 \le t \le 24$).

2. Curvature and optimisation: Use the first derivative to calculate and classify the peaks and troughs of daily consumption ($D'(t) = 0$). Find the second derivative $D''(t)$, study the intervals of concavity/convexity, and determine the inflection points, explaining what they represent regarding the acceleration rate of consumption.

3. Integral and proposal: Calculate the total energy consumed during the day using the definite integral $E = \int_{0}^{24} D(t)\,dt$. Write a brief report (one page) that includes the resulting graph and a justified time-of-use pricing proposal (off-peak and peak hours) to encourage energy savings.

Assessment criteria:

• Rigour in analytically fitting the real data with GeoGebra or the spreadsheet.

• Accuracy and mathematical justification of extrema, curvature, inflection points, and integration.

• Contextualised interpretation of the physical meaning of the derivatives and feasibility of the final proposal.