Optimisation problems: notes and practice questions

A company is designing a closed cylindrical container to hold a specific volume of liquid. The total surface area of the container, in cm$^2$, with a fixed volume of $16\pi$ cm$^3$ and a radius of $r$ cm, is given by the function $A(r) = 2\pi r^2 + \frac{32\pi}{r}$, where $r > 0$.

Find $A'(r)$.

Solve $A'(r) = 0$.

Interpret your answer to (b)(i) in context.

A group of engineers is designing a new observation Ferris wheel. The height, $H(t)$, in metres, of a passenger capsule above the ground is modelled by the function $H(t) = p \cos\left(\frac{\pi}{100}t\right) + q$, where $t$ is the time in seconds after the capsule begins its ascent from the highest point.

The lowest point a capsule reaches is 2 metres above the ground, and the highest point is 32 metres above the ground. The Ferris wheel completes one full rotation in 200 seconds.

Find the values of $p$ and $q$.

Using your values from part (a), the function is $H(t) = 15 \cos\left(\frac{\pi}{100}t\right) + 17$.

(i) Find $H'(t)$.

(ii) Find $H''(t)$.

The engineers are particularly interested in the moment when the capsule's vertical speed is at its maximum, for the first time after $t=0$. This occurs at time $t=k$.

Calculate the value of $k$.

Calculate the height of the capsule at this time $k$.

A landscape architect is designing a modular planter box for a new urban garden. The planter box has a base and top that are identical sectors of a circle, each with radius $r$ cm and angle $\theta$ radians. The height of the planter box is $h = 2$ cm. The total length of metal frame used for all edges of the planter box is $L = 20$ cm. This includes two circular arcs, four radial edges for the top and bottom sectors, and three vertical connecting edges.

(a) Show that $r = \frac{7}{2+\theta}$.

(b) The planter box is designed to hold soil, enclosing a volume, $V$.

(i) Find an expression for $V$ in terms of $\theta$.

(ii) Find the expression for $\frac{dV}{d\theta}$.

(iii) Solve algebraically $\frac{dV}{d\theta} = 0$ to find the value of $\theta$ that will maximize the volume, $V$.

(a) A space probe, 'Voyager Alpha', is launched from a space station. The position of the probe at time $t$ days after launch is given by the vector

$\mathbf{r} = \begin{pmatrix} 10 \\ 20 \\ 5 \end{pmatrix} + t \begin{pmatrix} 80 \\ 40 \\ 10 \end{pmatrix}$

Distances are measured in thousands of kilometres.

Find the position vector of the probe $3$ days after launch.

(b) A second space probe, 'Explorer Beta', is launched from a different station. The position vector of this probe is given by the vector

$\mathbf{s} = \begin{pmatrix} -50 \\ -30 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 70 \\ 50 \\ 12 \end{pmatrix}$

Determine if the two flight paths intersect and, if so, state the point of intersection.

(c) The two probes were launched at the same time, so $\lambda = t$.

State, with a reason, whether the two probes actually collide.

(d) Calculate the distance between the two space stations (the initial launch points).

(e) Calculate the shortest distance that ever exists between the two probes and the time when this occurs. Assume $t \ge 0$ days.

A surveillance drone D is flying with a constant velocity, $\boldsymbol{v}$, measured in kilometres per hour, where

$\boldsymbol{v} = \begin{pmatrix} 8 \\ -6 \end{pmatrix}$.

At time $t = 0$ the drone is at a point A(50, 20) relative to an origin O, where distances are

measured in kilometres.

Find the position vector $\vec{OD}$ of the drone at time $t$ hours.

A protected bird's nest is located at a point N(99, 2).

Find the value of $t$ when the drone will be closest to the bird's nest.

An environmental sensor will trigger if the drone flies within 18 kilometres of the nest.

State whether the sensor will trigger. Give a reason for your answer.

A company is designing an open-top storage container with a square base. The side length of the base is $x$ cm and the height is $h$ cm. The container needs to have a volume of $500$ cm$^3$.

Explain why $x^2 h = 500$.

Rearrange the equation in part (a) to make $h$ the subject.

Write down an expression for the surface area, $A$, of the open-top container.

Show that this can be written as

$A = x^2 + \frac{2000}{x}$

Plot the graph of $A = x^2 + \frac{2000}{x}$ for $x > 0$.

Find the minimum surface area and the value of $x$ when this occurs.

A conical tank is being filled with water. The tank has a height of 12 cm and a radius of 4 cm at its top. Water is poured into the tank such that the volume of water is increasing at a rate of 10 cm$^3$s$^{-1}$.

(a) Find the height of the water in the tank when its volume is 8$\pi$ cm$^3$.

(b) Hence or otherwise, find the rate of change of the height of the water at this instant.

(a) A pharmaceutical company is designing a new cylindrical container for a special liquid. The container has a fixed volume $V = 500 \text{ cm}^3$.

The radius of the container is $r \text{ cm}$ and the height is $h \text{ cm}$.

Show that $\pi r^2 h = 500$.

(b) Find an expression for the total surface area $S$ of the container.

(c) Substitute an expression for $h$ (from part (a) ) into your expression for $S$ (from part (b) ) and hence show that $S = 2\pi r^2 + \frac{1000}{r}$.

(d) Find $\frac{dS}{dr}$.

(e) Find the minimum value of $S$ and the values of $r$ and $h$ when this occurs. Show that this value of $S$ is indeed a minimum.

A landscape architect is designing a series of modular planter boxes for an urban garden project. The volume, $V$ cm$^{3}$, of a particular planter box is modelled by the function $V(d) = 30d^2 - d^3$, where $d$ is the depth of the planter in cm.

(a) Use your graphic display calculator to find the value of $d$ that will produce the maximum volume.

(b) Show that the maximum volume of the planter box is $4000$ cm$^{3}$.

The width of the planter is $d$ cm, and the length is $(30 - d)$ cm.

(c) The architect is interested in the total linear dimension $L$, which is defined as the sum of the depth, width, and length of the planter. Hence find the value of $L$ when the volume is maximized.

A landscape architect is designing a new public park. The northern boundary of the park is modelled by the function $g(x)$, and other boundaries are straight lines. The plan view of the park is shown in the following diagram, where both axes represent distance and are measured in metres.

The function $g(x)$ models the northern boundary of the park between points B and C and is given by

$g(x) = -\frac{1}{10}x^2 + 4x + 15$, for $0 \le x \le 40$.

(i) Find $g'(x)$.

(ii) Hence find the coordinates of the point on the park's northern boundary that is furthest north.

Point A has coordinates $(0, 0)$, point B has coordinates $(0, 15)$, point C has coordinates $(40, 15)$ and point D has coordinates $(40, 0)$.

(i) Write down the integral which can be used to find the area of the shaded region representing the park.

(ii) Find the area of the park.

(i) A landscaper uses the trapezoidal rule with 4 intervals to estimate the area of the park. Calculate this estimate.

(ii) Calculate the percentage error in the landscaper's estimate.

(iii) Suggest how the landscaper might be able to reduce the error whilst still using the trapezoidal rule.

A square-shaped meditation garden, PQRS, is to be built within the park. The side [PS] lies on the southern boundary (x-axis), and the side [QR] lies on the eastern boundary of the park ($x=40$). Point Q lies on the northern boundary curve $g(x)$.

(i) Find the x-coordinate of point P for the largest area of the meditation garden.

(ii) Find the largest area of the meditation garden.

A company's profit, $P(x)$, in thousands of dollars, from selling $x$ thousand units of a new smart device, is modelled by the function $P(x) = \frac{2x+1}{x^2+3}$, for $x \ge 0$.

Find an expression for $P'(x)$.

Find the equation of the tangent to the profit curve at the point where $x = 1$.

Determine the number of units sold (to the nearest unit) that maximizes the company's profit, and state the maximum profit (to two decimal places).

Determine the range of values of $x$ (number of thousands of units sold) for which the company's profit is increasing.

(a) A confectionery company is designing a new wrapper for a chocolate bar. The wrapper is in the shape of a rectangular prism with a square base of side length $x$ cm. The height, $h$ cm, is twice the side length of the base.

Write down an expression for $h$ in terms of $x$.

(b) The chocolate bar has a volume of $250$ cm$^3$.

Find the value of $x$ and $h$.

(c) Calculate the total external surface area of this rectangular prism wrapper.

(d) The company also considers a cylindrical wrapper with radius $r$ cm and height $H$ cm. This wrapper must also hold $250$ cm$^3$ of chocolate.

Find an expression for the height, $H$, of the cylindrical wrapper in terms of $r$.

(e) Let the total external surface area of the cylindrical wrapper be $A$ cm$^2$.

Show that $A = 2\pi r^2 + \frac{500}{r}$.

(f) Find $\frac{dA}{dr}$.

(g.i) Hence or otherwise, find the value of $r$ that will minimize $A$.

(g.ii) Find the minimum value of $A$ for the cylindrical wrapper.

(h) To account for manufacturing waste and overlap, an additional $12\%$ of the calculated surface area is required for the rectangular prism wrapper, and $20\%$ for the cylindrical wrapper.

Determine which wrapper design the company should choose to minimize material usage. Justify your answer.

A construction company is designing a drainage channel with a rectangular cross-section. They have a total of $100$ metres of a special waterproof lining material to form the base and the two vertical sides of the channel's cross-section. Let $x$ represent the height of the channel in metres.

(a) Find an expression for the cross-sectional area, $A$, of the channel in terms of $x$, in its simplest form.

(b) Determine the value of $x$ that maximizes the cross-sectional area.

(c) Calculate the maximum cross-sectional area of the channel.

A marine engineer is modelling the vertical displacement, $h$ meters, of a buoy from its equilibrium position at time $t$ minutes after being disturbed by a wave. The motion is described by the differential equation:

$\frac{\text{d}^2 h}{\text{d}t^2} + 6\frac{\text{d}h}{\text{d}t} + 8h = 0$.

This equation can be rewritten as a system of coupled first-order differential equations:

$\frac{\text{d}h}{\text{d}t} = y$

$\frac{\text{d}y}{\text{d}t} = -8h - 6y$.

Find the general solution for $h$.

Initially, the buoy is at its equilibrium position ($h = 0$) and has an initial upward velocity of $2$ m/min ($\frac{\text{d}h}{\text{d}t} = 2$).

Find an expression for $h$ in terms of $t$.

Sketch $h$ against $t$ in the interval $0 \le t \le 4$.

The engineer is interested in the maximum upward displacement of the buoy.

Find the time, in minutes, when this maximum displacement occurs.

Calculate the maximum upward displacement of the buoy.

An improved model for the buoy's motion, considering an external periodic force, is given by:

$\frac{\text{d}^2 h}{\text{d}t^2} + 6\frac{\text{d}h}{\text{d}t} + 8h = h \cos t$.

The same initial conditions as in part (b.i) apply ($h(0) = 0$ and $\frac{\text{d}h}{\text{d}t}(0) = 2$).

Use Euler's method with a $t$-interval of $0.1$ to predict the value of $h$ when $t = 1$.

(a) Two drones, P and Q, are launched simultaneously. Drone P is launched from the origin $(0,0)$ of a coordinate system, while Drone Q is launched from a point $15$ units along the positive horizontal axis. The velocity equations for the two drones are given by $\mathbf{v}_P$ and $\mathbf{v}_Q$ where:

$\mathbf{v}_P = \begin{pmatrix} 6 \\ 4 - 2t \end{pmatrix}$ and $\mathbf{v}_Q = \begin{pmatrix} -3 \\ 6 - 2t \end{pmatrix}$

Write down vector equations for the displacement, $\mathbf{r}_P$ and $\mathbf{r}_Q$, of the two drones at time $t$.

(b) Find an expression for the vector joining the centre of Drone P to the centre of Drone Q.

(c) Hence find the shortest distance between the two drones.

(d) The pilot of Drone P adjusts its initial velocity vector to $\begin{pmatrix} a \\ b \end{pmatrix}$. The velocity equation for Drone P is now $\mathbf{v}_P = \begin{pmatrix} a \\ b - 2t \end{pmatrix}$.

Find the values of $a$ and $b$ if the two drones collide when $t = 1.5$ seconds.

(e) Using the values of $a$ and $b$ found in part (d), find the angle to the horizontal at which Drone P was initially launched.

(a) SweetTreats is designing new packaging for a line of artisanal chocolates. The initial design is a box in the shape of a cuboid with a square base of side length $L$ cm. Its height, $H$ cm, is twice the length of the base.

Write down an expression for $H$ in terms of $L$.

(b) The box is designed to hold $250$ cm$^3$ of chocolates.

Find the value of $L$ and $H$.

(c) Calculate the total external surface area of the box.

(d) To minimize the amount of material needed, SweetTreats is considering changing the shape to a cylinder with radius $r$ cm and height $h$ cm. The cylindrical container must also hold $250$ cm$^3$ of chocolates.

Find an expression for the height, $h$, of the container in terms of $r$.

(e) Let the total external surface area of the cylindrical container be $A$ cm$^2$.

Show that $A = 2\pi r^2 + \frac{500}{r}$.

(f) Find $\frac{\text{d}A}{\text{d}r}$.

(g.i) Hence or otherwise, find the value of $r$ that will minimize $A$.

(g.ii) Find the minimum value of $A$ needed for the cylinder.

(h.i) Find $\frac{\text{d}^2 A}{\text{d}r^2}$.

(h.ii) Hence determine whether the graph of $A$ is concave-up or concave-down for $r > 0$. Justify your answer.

(a) A builder is designing a rectangular enclosure for a new community garden. The total length of fencing available for the enclosure is $60$ metres.

The builder wants to maximize the area of the garden. Calculate the maximum possible area of the enclosure.

(b) To ensure structural stability, the builder also needs to consider the sum of the squares of the lengths of the sides. Calculate the minimum possible value for the sum of the squares of the lengths of the sides of the enclosure.

(c) Justify that the value found in part (b) is indeed a minimum.

The rate of change of pollution, $\frac{dP}{dt}$ (in tonnes per day), in a protected lake is modelled by $\frac{dP}{dt} = 1 - 0.1t$, where $t$ is the time in days since monitoring began, for $0 \le t \le 15$.

(i) Find the value of $\frac{dP}{dt}$ at $t = 4$ days.

(ii) Interpret the meaning of your answer to part (a) (i) in context.

Use $\frac{dP}{dt}$ to find the value of $t$ when the pollution in the lake reaches its maximum level.

Two days after monitoring began, the pollution in the lake was $5.5$ tonnes.

Find an expression for $P$ in terms of $t$, for $0 \le t \le 15$.

Hence, find the maximum pollution level in the lake.

A second source of pollution, from a nearby factory, is modelled by $Q = 3(1.15)^t$, where $Q$ is the pollution in tonnes and $t$ is the time in days, for $0 \le t \le 15$.

Write down the initial pollution from this factory.

Each day, the pollution from the factory increases by $p$ %.

Find the value of $p$.

Find the value of $t$ when the pollution from the initial lake source and the factory are the same.

A rectangular sheet of metal, measuring $50 \text{ cm}$ by $30 \text{ cm}$, has a square of side length $x \text{ cm}$ cut from each corner. The sides are then folded up to form an open-top box in the shape of a cuboid.

Show that the volume $V$ of the cuboid may be expressed as $V = 4x^3 - 160x^2 + 1500x$.

Find an expression for $\frac{dV}{dx}$.

Hence show that the cuboid will have a maximum volume when $3x^2 - 80x + 375 = 0$.

Using technology, find the maximum possible volume of the cuboid. Give your answer correct to one decimal place.

The 'EcoPack' company designs sustainable packaging. They produce a standard closed rectangular storage container with a length of $10$ cm, a width of $6$ cm, and a height of $4$ cm. The information is shown in the diagram.

Calculate the surface area of the container in cm$^2$.

(b) Calculate the length of the longest internal diagonal of the container.

(c) Each week, EcoPack sells $x$ thousand containers. It is known that $\frac{dP}{dx} = -3x+300$, for $x \ge 0$, where $P$ is the weekly profit, in dollars, from the sale of $x$ thousand containers.

Find the number of containers that should be sold each week to maximize the profit.

(d) The profit from the sale of $30000$ containers is $2500.

Find $P(x)$.

(e) Find the least number of containers which must be sold each week in order to make a profit.