Function Optimization
Wilkin has 80 meters of fencing and wants to enclose a rectangular pen for his chickens, with as much space as possible. What dimensions should he choose? And what if the fencing cost more on one side? Every day, in a business, on a farm, or in a factory, someone asks how to spend as little as possible or earn as much as possible. The derivative solves that kind of question.
The five-step procedure
An optimization problem asks for the largest or the smallest value of a quantity (area, cost, time, profit). It is always solved the same way.
Let's go back to Wilkin's pen, but with 80 m of fencing that goes around all four sides.
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,…
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.
Divide the number $12$ into two positive parts such that the sum of their squares is minimum.
Divide the number $12$ into two positive parts such that the sum of their squares is minimum.
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…
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.
When the function is already given
Sometimes the problem gives you the function: the height of a probe, the profits of a business, the "likes" on a post. Then you jump straight to step 4: take the derivative, set it equal to zero, and check.
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…
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…
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 $B(x) = -120x^2 + 1\,920x - 2\,400$.
Find the number of units that must be manufactured each month to maximize profit, and calculate the value of this maximum profit.
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…
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.
Rectangles: area and perimeter
In problems with plane figures, the constraint relates the two sides ($2x+2y=P$ or $xy=A$). You solve for one side and get a function of one variable.
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…
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?
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…
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 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…
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.
Open-top boxes
A square of side $x$ is cut from each corner of a square sheet and the edges are folded up. It is a classic: the volume rises and then falls, so there is a maximum.
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…
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 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…
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.
Price and revenue models
In many businesses, raising the price reduces sales. Where is the balance that brings in the most money? It is modeled with a quadratic function in the variable "number of increases."
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…
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?
Cylinders and fixed volume
Containers (cans, tanks, capsules) usually have a fixed volume, and the goal is to use the minimum material. The constraint is the volume formula.
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 $300$…
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 $300$ pesos per square meter, while the material for the curved surface is lighter and costs $120$ 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.
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…
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.
Costs with different materials or sections
When each part of a figure has a different price, the total cost is the sum of each part times its price.
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…
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 $\text{RD\$}\ 7\,200$ per linear meter, while the internal wall, being less strong, costs $\text{RD\$}\ 4\,800$ per linear meter. Find the dimensions of the warehouse that make the cost of the walls as low as possible.
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 $1.20 \text{ \text{RD\$}\…
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 $1.20 \text{ \text{RD\$}\ /cm}^2$, while the material for the side costs $0.60 \text{ \text{RD\$}\ /cm}^2$. 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 $3 \text{ \text{RD\$}\ /cm}$ for the strip. Calculate the radius and height of the tin that minimise the total production cost.
Minimum distances
To find the point on a curve closest to another point, you minimize the distance. Trick: the distance is minimal where its square is, and that avoids the square root.
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 $y = x^2$.…
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.
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…
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.
Inscribed figures
When one figure is inside another (a rectangle in a circle, in a triangle, a beam in a log), the constraint comes from geometry: the Pythagorean theorem or similar triangles.
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…
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.
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…
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 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…
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.
Endpoints of the interval and cases with no solution
When the variable is in a closed interval, the maximum or the minimum can be at an endpoint. And when the interval is open, it may happen that there is no maximum or minimum.
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…
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.
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,…
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.