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.

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 clothing store sells $80$ T-shirts each week for $\$12$ each. It estimates that for every pesos 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 pesos.

1. b) Find the price that maximizes 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 municipality 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 minimize the required land, as well as the minimum total area.

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 $x$ meters 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 minimizes $L(x)$. (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.)