Which rectangle with a fixed perimeter has the largest area? Which dimensions make a box use the least cardboard? These questions have something in common: there is a quantity that can be made larger or smaller by choosing one measurement well, and we have to decide whether an optimal value exists and where it is reached. There may be several optimal points, or the limiting value may never be reached: that is why the domain is part of the problem.
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.
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.
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.
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.
The most common type in exams: a shape (rectangle, pen, box) with one fixed measurement (perimeter, amount of fencing, surface area of cardboard), and you are asked for the shape that makes another quantity (area, volume, cost) as large or as small as possible.
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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 a perimeter of $36\ \text{m}$, find the dimensions of the one with maximum area and calculate that area.
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…
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.
a) Express the area of the vegetable patch as a function of the side $x$ perpendicular to the wall.
b) Find the dimensions that maximise the area and the value of that area.
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…
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.
a) Express the volume of the box $V(x)$ as a function of $x$ and state the valid domain for $x$.
b) Find the value of $x$ that maximises the volume.
c) Calculate the maximum volume and check, using the sign of $V'(x)$ on both sides, that it is indeed a maximum.
Not all problems involve geometric shapes: the same procedure works to minimise a cost, to maximise a volume with a limit on the material, or to find the extremum of a model that depends on time.
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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. a)…
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.
a) Express the total area of the pen as a function of one of its sides.
b) Determine the dimensions that maximise the total area and calculate that area.
A clothes shop sells $80$ T-shirts each week for $12\ \text{€}$ each. It estimates that for every euro it reduces the price, it will sell $10$ more T-shirts per week. a)…
A clothes shop sells $80$ T-shirts each week for $12\ \text{€}$ each. It estimates that for every euro it reduces the price, it will sell $10$ more T-shirts per week.
a) Express the weekly revenue $I(x)$ as a function of the price reduction $x$ in euros.
b) Find the price that maximises the revenue and the maximum weekly revenue.
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…
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: $R = k \cdot b \cdot h^2$.
a) Express the strength as a function of the base $b$ using the relationship connecting $b$, $h$ and $d$.
b) Find the dimensions of the beam with maximum strength.
A rectangle with perimeter $p$ has area $A(x)=x(p/2-x)$, with $0
Communicate your initial prediction, the measurements from the graph and the proof. Explain which part is contributed by the tool and which part by the reasoning.
Let's practice! 10 medium exercises in this section
Back to the beginning: Of all the rectangles with the same perimeter, the one with the greatest area is always the square; and for a box with a square base and a fixed volume, the minimum use of cardboard is achieved by making the box in the shape of a cube.
Summary
Let's practice! 10 medium exercises in this section