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 $120$ meters 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 maximized.

A small weather probe is launched vertically upward. The height $h$ (in meters) 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 pesos) from the sale of a new model of wireless headphones depends on the number of units manufactured $x$ (in hundreds) according to the function $B(x) = -40x^2 + 640x - 800$.

Find the number of units that must be manufactured each month to maximize 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.

An animal shelter has received a donation of $120$ meters 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 maximized.

A streaming platform is analyzing its pricing policy for a new premium plan. Currently, they have $10,000$ subscribers who pay $12$ pesos per month. Market research indicates that for every pesos the subscription price is reduced, $1,000$ new users will join. Conversely, for every pesos the price increases, $1,000$ users will be lost. Knowing that the fixed maintenance costs are $20,000$ pesos per month and that each subscriber generates an administrative cost of $2$ pesos, what monthly subscription price will maximize 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 meters. The material used for the bases (top and bottom) costs $100$ pesos per square meter, while the material for the curved surface is lighter and costs $40$ pesos per square meter. Calculate the radius of the base and the height of the cylinder that minimize the total manufacturing cost of each tank.

An advertising board shaped like an isosceles triangle inscribed in a semicircle of radius $R = 6$ meters is to be designed, so that its base coincides with the diameter of the semicircle. Determine the dimensions of the triangle (base and height) so that the area of the advertising surface is maximized.

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 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 $100 \text{ km/h}$. Calculate the exact point on the dividing line that the drone must cross so that the total travel time is minimized.

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 $\$2\,400$ per linear meter, while the internal wall, being less strong, costs $\$1\,600$ per linear meter. 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 $0.40 \text{ \$/cm}^2$, while the material for the side costs $0.20 \text{ \$/cm}^2$. 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 $1 \text{ \$/cm}$ 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 $20 \text{ meters}$. 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 maximized? And so that it is minimized? Justify your answer by analyzing the endpoints of the interval of definition of the variable.