A coffee roaster in Seattle wants a cylindrical can that holds exactly $54\pi$ cubic inches and uses as little metal as possible. Should the can be tall and skinny, short and wide, or something in between? Calculus answers this exactly. Every optimization problem turns a real question into a function of one variable and then looks for where its derivative is zero.
The five-step method
The five steps below solve almost every optimization problem.
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Determine two positive real numbers whose sum is exactly $60$ and such that their product is as large as possible.
Determine two positive real numbers whose sum is exactly $60$ and such that their product is as large as possible.
Sometimes the problem hands you the function, such as the height of a projectile, a profit formula, or a count of likes. Then steps 1–3 are done and you go straight to critical points. If the time or quantity is restricted to an interval like $0\le t\le 10$, the interval is closed and you must use the Candidates Test.
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A small weather probe is launched vertically upward. The height $h$ (in feet) 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 feet) 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 instant 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 dollars) 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 dollars) 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 maximize the profit and calculate the value of this maximum profit.
The number of "likes" on a post by a famous influencer during the first $10$ hours follows the function $L(t) = -t^3 + 12t^2 + 45t$, where $t$ is the time elapsed in…
The number of "likes" on a post by a famous influencer during the first $10$ hours follows the function $L(t) = -t^3 + 12t^2 + 45t$, where $t$ is the time elapsed in hours ($0 \le t \le 10$). Determine at what time the post reaches its greatest success (maximum number of likes) and how many likes it gets at that instant.
Sometimes the derivative gives only the minimum and the maximum is at the edge of the domain. Cutting a string into two shapes is a classic case.
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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 of them 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 of them and an equilateral triangle with the other. How should we make the cut so that the sum of the areas of both figures is a maximum? And so that it is a minimum? Justify the answer by analyzing the endpoints of the interval of definition of the variable.
In fence problems, the secret is to count the segments of fence. Draw the picture and label every piece.
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A young farmer has $120$ feet of fencing to enclose a rectangular grazing area. Taking advantage of the fact that one of the sides of the enclosure will be an existing…
A young farmer has $120$ feet of fencing to enclose a rectangular grazing area. Taking advantage of the fact that one of the sides 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 a maximum.
An animal shelter has received a donation of $120$ feet of wire mesh to build two adjoining rectangular enclosures for dogs and cats. The enclosures must share a common…
An animal shelter has received a donation of $120$ feet of wire mesh to build two adjoining rectangular enclosures for dogs and cats. The enclosures must share a common side, as shown in the figure, to make the most of the material. Determine the dimensions of the outer structure ($x$ and $y$) so that the total enclosed area is a maximum.
When the area is fixed, the constraint is $xy=A$, and so $y=A/x$. The domain is an open interval $(0,\infty)$. Check what happens at the ends: if the objective goes to $\infty$ at both ends, then the only critical point must be an absolute minimum.
Page-margin problems have one more idea. The page has total area $wh$, but the printed area is smaller. It is $(w-2m_s)(h-2m_t)$, where $m_s$ is the side margin and $m_t$ the top and bottom margin.
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A graphic design studio wants to create a rectangular greeting card containing exactly $32 \text{ cm}^2$ of printed surface. If margins are not considered, what should…
A graphic design studio wants to create a rectangular greeting card containing exactly $32 \text{ cm}^2$ of printed surface. If margins are not considered, what should the dimensions of the card be so that the perimeter is a minimum and thus save on paper cutting?
A page of an art book must have a total area of $600$ $cm^2$. The top and bottom margins must be $3$ in each, and the side margins $2$ in 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 $3$ in each, and the side margins $2$ in each. Determine the dimensions of the page (width and height) so that the area of the printed zone (where the text and images go) is a maximum.
To fit a rectangle inside a triangle or a circle, you need an equation linking the variables. Two tools do almost everything:
Similar triangles relate the pieces when a figure sits inside a triangle.
The Pythagorean Theorem relates the pieces when a vertex lies on a circle. A rectangle inscribed in a circle has its diagonal equal to the diameter.
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An advertising panel is to be designed in the shape of an isosceles triangle inscribed in a semicircle of radius $R = 6$ feet, so that its base coincides with the…
An advertising panel is to be designed in the shape of an isosceles triangle inscribed in a semicircle of radius $R = 6$ feet, 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 a maximum.
In a metalworking shop, a steel beam with a rectangular cross-section is to be cut from a cylindrical log of $40$ in diameter. The mechanical resistance $R$ of this beam…
In a metalworking shop, a steel beam with a rectangular cross-section is to be cut from a cylindrical log of $40$ in diameter. The mechanical resistance $R$ of this beam is proportional to the product of the width $x$ and the square of the height $y$, that is, $R = k \cdot x \cdot y^2$ (where $k$ is a resistance constant of the material). Find the dimensions $x$ and $y$ that the beam must have so that its resistance is a 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…
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 upper 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, so that the total energy generated is given by the function $E = A \cdot (5 - h)$, where $A$ is the area of the panel and $h$ is its height. Find the dimensions of the panel that maximize the energy generated.
Cut a square from each corner of a square sheet and fold up the sides. The cut size $x$ is the height of the box. This is a classic problem, and it is a closed-interval problem: $x$ can be as small as $0$ and as large as half the side.
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An open box (without a lid) is to be built from a square sheet of cardboard of side $30 \text{ cm}$. To do this, a square of side $x$ is cut from each of the corners and…
An open box (without a lid) is to be built from a square sheet of cardboard of side $30 \text{ cm}$. To do this, a square of side $x$ is cut from each of the corners and the edges are folded up. Express the volume of the box as a function of $x$ and determine the value of $x$ so that the volume of the box is a maximum.
A graphic design studio wants to create a promotional packaging box from a square piece of cardboard of side $24$ in. To do this, a square of side $x$ is cut from each…
A graphic design studio wants to create a promotional packaging box from a square piece of cardboard of side $24$ in. To do this, a square of side $x$ is cut from each of the corners and the flaps are folded up to form an open box. Find the value of $x$ so that the capacity of the box is as large as possible.
In 3D problems the volume is usually fixed. Solve the volume formula for one variable (usually $h$), then substitute into the surface area, which is the amount of material.
Back to the opening question: the cheapest can is neither tall and skinny nor short and wide. Its height $6$ in. is exactly its diameter $2r=6$ in. In fact, with equal material costs, the optimal closed cylinder always has $h=2r$.
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A chemical company needs to manufacture closed cylindrical tanks with a capacity of $16 \pi$ cubic feet. The material used for the bases (top and bottom) costs $\$5…
A chemical company needs to manufacture closed cylindrical tanks with a capacity of $16 \pi$ cubic feet. The material used for the bases (top and bottom) costs $\$5$ per square foot, while the material for the lateral surface is lighter and costs $\$2$ per square foot. Calculate the base radius and the height of the cylinder that minimize the total manufacturing cost of each tank.
When parts of an object use different materials or have different prices, the objective is a cost, and you add the cost of each part separately.
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A logistics company is building a warehouse with an area of $400 \text{ ft}^2$. The warehouse must have a rectangular shape and be divided into two sectors by an…
A logistics company is building a warehouse with an area of $400 \text{ ft}^2$. The warehouse must have a rectangular shape and be divided into two sectors by an internal wall parallel to one of its lateral sides. The cost of the external walls is $\$120 \text{ \$}$ per linear foot, while the internal wall, being less resistant, costs $\$80 \text{ \$}$ per linear foot. Find the dimensions of the warehouse that make the cost of the walls as low as possible.
A tin can manufacturer wants to create a new cylindrical container with a volume of $500 \text{ cm}^3$. The material for the lid and the base costs…
A tin can manufacturer wants to create a new cylindrical container with a volume of $500 \text{ cm}^3$. The material for the lid and the base costs $0,02 \text{ \$/cm}^2$, while the material for the side costs $0,01 \text{ \$/cm}^2$. In addition, to reinforce the structure, a metal strip must be placed around the joint of the lid with the side and another around the base, with a cost of $0,05 \text{ \$/cm}$ of strip. Calculate the radius and height of the can that minimize the total production cost.
We want to design a capsule for a new drug that has the shape of a right circular cylinder extended at its ends by two hemispheres of the same radius as the cylinder.…
We want to design a capsule for a new drug that has 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 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 minimize the total manufacturing cost.
Business problems use the relation Profit = Revenue − Cost. With a linear demand model, the number sold changes by a fixed amount for each dollar of price change. Choose the price $x$ as the variable and write the quantity sold as a linear function of $x$.
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A streaming platform is analyzing its pricing policy for a new premium plan. Currently, they have $10.000$ subscribers paying $\$12…
A streaming platform is analyzing its pricing policy for a new premium plan. Currently, they have $10.000$ subscribers paying $\$12$ a month. A market study indicates that for every dollar the subscription price is reduced, $1.000$ new users will sign up. Conversely, for every dollar the price increases, $1.000$ users will be lost. Knowing that the fixed maintenance costs are $\$20.000$ a month and that each subscriber generates a management cost of $\$2$, what monthly subscription price will maximize the net profit of the platform?
Distance, tangent lines, and "prove that" problems
To find the point on a curve closest to a given point, write the distance with the distance formula. Then minimize the square of the distance.
Tangent lines also give optimization problems. Use the point-slope form $y-f(a)=f'(a)(x-a)$ and find where the line crosses each axis.
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In a strategy video game, a scouting unit is located at point $A(0, 5)$ on a coordinate map. There is a path whose trajectory follows the shape of the parabola…
In a strategy video game, a scouting 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.
Consider the function $f(x) = \frac{1}{x^2}$ for $x > 0$. A tangent line is drawn to the graph of this function 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 is drawn to the graph of this function 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 the value of $a$ for which the area of this triangle is a minimum and calculate the value of this area.
Two points $A$ and $B$ are located on the same side of a line $r$. Point $A$ is at a distance of $3 \text{ km}$ from the line and point $B$ is at $6 \text{ km}$. The…
Two points $A$ and $B$ are located on the same side of a line $r$. Point $A$ is at a distance of $3 \text{ km}$ from the line and 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 want 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 a minimum. Prove that this point coincides with the midpoint of the projection of $AB$ onto the line.
If a path crosses two regions with different speeds, the fastest route is usually not the straight line. The objective is a sum of times, and each time is distance divided by speed.
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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 miles. The terrain is…
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 miles. 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 $60 \text{ mph}$. In the lower zone ($y \leq 2$), the drone can reach $100 \text{ mph}$. Calculate the exact point on the dividing line where the drone must cross so that the total travel time is a minimum.