Exercises: Optimisation problems

Split the number $24$ into the sum of two positive numbers such that their product is maximum. Set up the function to be optimised and solve it using differentiation.

Among all rectangles with a perimeter of $36\ \text{m}$, find the dimensions of the one with maximum area and calculate that area.

Among all rectangles with an area of $100\ \text{m}^2$, find the dimensions of the one with minimum perimeter and the value of that perimeter.

The area of a rectangular pen built against a wall using $40\ \text{m}$ of fencing for the other three sides is given, as a function of one of its sides $x$, by $A(x) = 40x - 2x^2$. Find the value of $x$ that maximises the area and the value of this maximum area.

Which positive number exceeds its square by the greatest possible amount? Express this amount as a function of the number and maximise it using differentiation.

A gardener wants to fence off a rectangular vegetable patch against a wall of his house, so that he only needs to fence three sides. He has $60\ \text{m}$ of fencing available.

1. a) Express the area of the vegetable patch as a function of the side $x$ perpendicular to the wall.

1. b) Find the dimensions that maximise the area and the value of that area.

Find two positive numbers that sum to $18$ and whose sum of squares is minimum.

Reason whether the following statement is true or false: "The maximum value of a continuous function on a closed interval is always reached at a point where $f'(x) = 0$." If false, give a counterexample using a simple function on a specific interval.

Aoife wants to design a rectangular vegetable garden in her village by taking advantage of an existing stone wall, so that she only needs to fence three sides (two for width and one for length). She has $60\text{ m}$ of wire mesh and wants the cultivable area to be maximal. In her exam, Aoife writes:

«I call $x$ and $y$ the sides of the rectangle. Since the perimeter is $2x + 2y$, we have $2x + 2y = 60 \implies y = 30 - x$. The area function is $A(x) = x(30 - x) = 30x - x^2$. Differentiating: $A'(x) = 30 - 2x = 0 \implies x = 15\text{ m}$. Therefore, the optimal dimensions are $15\text{ m} \times 15\text{ m}$ and the maximum area is $225\text{ m}^2$».

1. a) Locate and explain the two conceptual errors made by Aoife: one in the geometric formulation of the constraint and another in the justification of the extremum.
2. b) Correct the exercise: define the chosen variable and its admissible physical domain, formulate the correct area function, justify using the derivative that it is an absolute maximum, and calculate the true dimensions and the maximum area of the garden.

An agricultural cooperative in Galway has a budget that allows it to install exactly $80\text{ m}$ of fencing to enclose a rectangular area intended for organic poultry farming.

1. a) Before carrying out any analytical calculation with derivatives, estimate with reasoning which rectangular geometric shape will provide the greatest possible area and venture what dimensions ($x$ and $y$) and what maximum area it should have. Justify your intuition based on the principle of balanced distribution and geometric symmetry.
2. b) Model the problem by defining the variable $x$ (length of one of the sides), its admissible domain in the physical context, and the area function $A(x)$.
3. c) Calculate the candidate for the optimum using $A'(x) = 0$, demonstrate using the sign of the first derivative or the second derivative that it is an absolute maximum, and calculate the maximum area. Does the exact result match your initial estimate? Explain why.

An open-topped box is to be made from a square piece of cardboard with sides of $24\ \text{cm}$ by cutting out four equal squares of side length $x$ from the corners and folding up the sides, as shown in the figure.

1. a) Express the volume of the box $V(x)$ as a function of $x$ and state the valid domain for $x$.

1. b) Find the value of $x$ that maximises the volume.

1. c) Calculate the maximum volume and check, using the sign of $V'(x)$ on both sides, that it is indeed a maximum.

A farmer has $240\ \text{m}$ of fencing to build a rectangular pen divided into two plots by an internal fence parallel to one of the sides, as shown in the figure.

1. a) Express the total area of the pen as a function of one of its sides.

1. b) Determine the dimensions that maximise the total area and calculate that area.

A canning company designs cylindrical tins with lids having a capacity of $500\ \text{cm}^3$, like the one in the figure.

1. a) Express the total surface area of sheet metal $S(r)$ as a function of the radius $r$ of the base, using the volume condition to express height in terms of radius.

1. b) Find the radius that minimises the amount of sheet metal (giving its exact value and an approximation rounded to two decimal places is sufficient).

1. c) What is the relationship between height and radius in the optimal tin?

A clothes shop sells $80$ T-shirts each week for $\text{€}\ 12$ each. It estimates that for every euro it reduces the price, it will sell $10$ more T-shirts per week.

1. a) Express the weekly revenue $I(x)$ as a function of the price reduction $x$ in euros.

1. b) Find the price that maximises the revenue and the maximum weekly revenue.

Find the point on the parabola $y = x^2$ closest to the point $P(0, 3)$.

1. a) Express the square of the distance from a general point on the parabola to $P$ (minimising the square of the distance is equivalent to minimising the distance).

1. b) Find the points on the parabola that minimise this distance and calculate the minimum distance.

A designer layouts a poster with a printed area of $400\ \text{cm}^2$ and margins of $4\ \text{cm}$ on all four sides.

1. a) Express the total area of the paper as a function of the width $x$ of the printed area.

1. b) Find the dimensions of the complete sheet that minimise paper usage.

A wire of length $12\ \text{m}$ is cut into two pieces: one piece forms a square and the other forms a circle, as shown in the figure.

1. a) If $x$ is the length of wire used for the square, express the total enclosed area $A(x)$.

1. b) Find the value of $x$ that minimises the total area (leave the result in terms of $\pi$).

1. c) What area is obtained if the wire is not cut and is used entirely for the circle? Is it greater or less than the minimum?

A local council is going to build a rectangular swimming pool of $200\ \text{m}^2$ surrounded by a path of width $2\ \text{m}$ on two opposite sides and $1\ \text{m}$ on the other two, as shown in the figure.

1. a) Express the total area of the land (swimming pool plus path) as a function of one side $x$ of the pool.

1. b) Find the dimensions of the pool that minimise the required land, as well as the minimum total area.

A passenger transport company operates an intercity bus route between Cork and Limerick. Currently, the ticket costs $10\text{ €}$ and an average of $400$ users travel per day. According to a mobility study based on surveys by the Central Statistics Office (CSO), for every reduction of $0.50\text{ €}$ in the ticket price, the number of daily passengers increases by $40$. Due to fleet constraints, the maximum capacity cannot exceed $640$ daily passengers. The company's daily costs consist of a fixed maintenance cost of $800\text{ €}$ and a variable cost of $2\text{ €}$ per transported passenger.

1. a) Model the daily profit function $B(p)$ in terms of the ticket price $p$. Justify in detail the admissible domain of $p$ taking into account the maximum capacity of the fleet.
2. b) Find the critical points of $B(p)$, determine the optimal ticket price and the corresponding number of passengers, and calculate the maximum daily profit.
3. c) Check whether the optimum lies in the interior of the admissible domain or at an endpoint of the interval, and classify it using the second derivative.

A winegrower from La Rioja plans to fence a rectangular plot of $500\text{ m}^2$ to grow two grape varieties. To separate them, they will erect an interior dividing fence parallel to one of the two exterior sides of the plot.

Fencing the exterior perimeter costs $16\text{ \euro}$ per linear metre, while the interior dividing fence, being made of a lighter mesh, has an unknown unit cost of $c\text{ \euro}$ per linear metre.

After commissioning a study to minimize the total expenditure, the technical team informs them that the minimum total cost is achieved when the length of the interior dividing fence measures exactly $20\text{ m}$.

1. a) Formulate the total cost function $C(y)$ in terms of the length $y$ of the interior fence, indicating its admissible domain.
2. b) From the given known optimum ($y = 20\text{ m}$), deduce the value of the unit cost $c$ of the dividing fence.
3. c) Check, using the second derivative, that for said value of $c$ the function indeed has a minimum, and calculate the minimum total cost of the installation.

Squares of side $x$ are cut out from the corners of a rectangular sheet of $60\ \text{cm} \times 30\ \text{cm}$ to construct an open box.

1. a) Express the volume $V(x)$ and state the valid domain of $x$.

1. b) Find the values of $x$ that make $V'(x)$ zero and discard the one that does not belong to the domain.

1. c) Verify that the valid value gives the maximum volume and calculate it.

A window consists of a rectangle topped by a semicircle, as shown in the figure. The total perimeter of the window is $6\ \text{m}$.

1. a) Express the total area as a function of the radius $r$ of the semicircle.

1. b) Find the value of $r$ that maximises the light entering (leave the result in terms of $\pi$ and give an approximation to two decimal places).

1. c) What is the relationship then between the height of the rectangle and the radius?

A beam of rectangular cross-section is to be cut from a log of circular cross-section of diameter $d$, as shown in the figure. The bending strength of the beam is proportional to the product of its base and the square of its height: $R = k \times b \times h^2$.

1. a) Express the strength as a function of the base $b$ using the relationship connecting $b$, $h$ and $d$.

1. b) Find the dimensions of the beam with maximum strength.

A rectangle is inscribed with its base on the x-axis under the parabola $y = 4 - x^2$, as shown in the figure.

1. a) Express the area of the rectangle as a function of the x-coordinate $x$ of its top right vertex.

1. b) Find the dimensions of the rectangle of maximum area and the value of this area.

A cone whose slant height is $9\ \text{cm}$ is formed from a circular sector of card of radius $9\ \text{cm}$, as shown in the diagram.

1. a) Express the volume of the cone as a function of its height $h$ using the relationship between slant height, radius and height.

1. b) Find the height of the cone of maximum volume.

1. c) Calculate the radius of the base of this cone.

For the problem of the open box constructed from a square piece of card of side $32\ \text{cm}$, whose volume is $V(x) = x \times (32 - 2x)^2$ with $0 < x < 16$, a student writes: "$V'(x) = 12x^2 - 256x + 1024 = 0$ gives $x = 16$ and $x = \frac{16}{3}$; since volume increases with the size of the cut-out, I choose $x = 16\ \text{cm}$". Point out the two mistakes made in the conclusion (one concerning domain and one concerning justification) and solve the problem correctly.

Prove, using calculus, the following general result: of all rectangles with perimeter $4p$, the one with maximum area is the square of side $p$. Set up the area function in terms of one of the sides, justify that the critical value is a maximum, and conclude.

Two poles of height $6\ \text{m}$ and $8\ \text{m}$ are $10\ \text{m}$ apart on level ground. A taut cable is to be anchored from the top of each pole to the same point on the ground between them.

1. a) If the anchor point is $x$ metres from the base of the pole of height $6\ \text{m}$, express the total length of the cable $L(x)$.

1. b) Find the value of $x$ that minimises $L(x)$. (Hint: by setting $L'(x) = 0$, you will reach a proportion between legs and hypotenuses; solve it without expanding unnecessarily.)

1. c) Calculate the minimum length of the cable. (Hint: it coincides with the straight-line distance between the top of one pole and the reflection of the top of the other pole across the ground.)

We want to determine the point on the curve $y = x^2$ whose distance to the point $P(0;\, 2)$ is minimal. To address the problem, three students propose different strategies:

• Seán sets up the Euclidean distance function $d(x) = \sqrt{x^2 + (x^2 - 2)^2}$ and prepares to calculate $d'(x)$ directly with the chain rule.

• Aoife proposes minimizing the squared distance function: $S(x) = [d(x)]^2 = x^2 + (x^2 - 2)^2$, stating that both functions reach their extrema at the same points and that this approach significantly reduces algebraic complexity.

• Cian proposes a geometric method: at the closest point $Q(x;\, x^2)$, the vector $\vec{PQ}$ must be perpendicular to the tangent line to the parabola.

1. a) Formally justify why Aoife's proposal is mathematically rigorous, indicating which property of the auxiliary function $g(t) = t^2$ on $[0;\, +\infty)$ guarantees that the extrema coincide.
2. b) Solve the problem applying Aoife's method: find the candidates for extrema, classify them by analyzing the sign of $S''(x)$ or $S'(x)$, and calculate the coordinates of the closest points and the minimum distance.
3. c) Apply Cian's geometric method to check the obtained points. Compare the three methods, analyzing which one turns out to be more efficient and carries less risk of operational errors.

EirGrid is planning to lay a high-voltage cable to connect an offshore wind farm in Galway Bay to a coastal substation $S$. The offshore platform $P$ is located $6\text{ km}$ out to sea, perpendicular to a point on the coast $B$. The substation $S$ is located on the straight coastline at a distance $L$ from $B$.

Laying the submarine cable costs $50\,000\text{ €/km}$ and laying it on land (along the coast from a landing point $X$ to the substation $S$) costs $30\,000\text{ €/km}$. The cable will run in a straight line by sea from $P$ to $X$ (with $X$ located on the coastal segment between $B$ and $S$), and from $X$ to $S$ by land.

1. a) If the substation is at a distance $L = 10\text{ km}$ from $B$, define the variable $x$ as the distance from $B$ to the landing point $X$, establish its admissible domain, and write the total cost function $C(x)$ in thousands of euros.
2. b) Solve $C'(x) = 0$, compare the obtained value with the endpoints of the interval $[0;\, 10]$, and determine the optimal location of the landing point $X$ and the minimum cost of the cable laying.
3. c) Now assume that the substation were located only $L = 3\text{ km}$ from $B$. Explain why the critical point calculated in the previous section is no longer admissible, analyze the sign of $C'(x)$ in the new domain $[0;\, 3]$, and determine with justification where the cable should come ashore to minimize the cost.

Unit challenge. At what rate does the temperature change in your city? Become a meteorological analyst to model the thermal evolution of your locality applying differential calculus.

1. Real data and ARC: Go to the official Met Éireann website, record the temperature of your municipality every $3$ hours over a full day ($8$ records), and calculate the Average Rate of Change between the early morning and early afternoon. Explain its physical interpretation ($^\circ\text{C}/\text{h}$).

2. Modelling and derivative: Enter the data into GeoGebra or a spreadsheet to obtain a polynomial fit function $T(t)$ of degree $3$. Analytically apply differentiation rules to find the derivative function $T'(t)$. Obtain the equation of the tangent line at $t = 12.0\text{ h}$ and graphically interpret its slope.

3. Optimization and conclusions: Solve the equation $T'(t) = 0$ to locate relative extrema (theoretical maximum and minimum) and study the intervals of increase and decrease. Write a brief report comparing the predicted optimal time and value with the observed real data.

Assessment criteria:

• Rigorous collection of Met Éireann data and correct calculation and physical interpretation of the ARC.

• Precision in the algebraic calculation of $T'(t)$ and the tangent line.

• Proficient application of the derivative for optimization (extrema and monotonicity) and critical thinking when comparing the model with reality.