Optimisation: notes and practice questions

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis.

(a) Write down the coordinates of the points where the arch meets the ground and the highest point of the arch.

(b) Find the equation of the parabolic arch.

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis. A rectangular banner is placed underneath the arch with its base on the ground. Let one of the vertices of the banner in the first quadrant be $(x, 0)$.

(c) Express the width and height of the rectangular banner in terms of $x$.

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis. A rectangular banner is placed underneath the arch with its base on the ground. Let one of the vertices of the banner in the first quadrant be $(x, 0)$.

(d) Write down an expression for the area of the rectangular banner, $A(x)$, in terms of $x$.

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis. A rectangular banner is placed underneath the arch with its base on the ground. Let one of the vertices of the banner in the first quadrant be $(x, 0)$.

(e) Find the value of $x$ for which the area of the banner is maximized.

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis. A rectangular banner is placed underneath the arch with its base on the ground. Let one of the vertices of the banner in the first quadrant be $(x, 0)$.

(f) Calculate the dimensions (width and height) of the banner that yield the maximum area.

A designer is creating a large decorative arch for a garden entrance. The arch has the shape of a parabola. Its base rests on the ground (the x-axis) and spans a width of $10 \text{ m}$. The maximum height of the arch is $8 \text{ m}$. The arch is symmetrical about the y-axis. A rectangular banner is placed underneath the arch with its base on the ground. Let one of the vertices of the banner in the first quadrant be $(x, 0)$.

(g) Find the maximum possible area of the inscribed rectangular banner.

This question asks you to investigate the motion of a buoy bobbing up and down in the water.

A buoy bobs up and down in the water.

A fixed origin $\text{O}$ is the equilibrium position of the buoy (the water level).

The buoy's displacement, $y$ metres, from $\text{O}$ at time $t$ seconds is given by

$y = 6\sin\left(2t + \frac{\pi}{6}\right), \text{ for } 0 \le t \le \pi.$

Determine

the amplitude of the buoy's motion;

the buoy's initial displacement from $\text{O}$;

the value of $t$ when the buoy first passes through $\text{O}$.

Now consider the general case of a buoy bobbing up and down.

The buoy's acceleration is always directed towards a fixed origin $\text{O}$ at its equilibrium position.

The buoy's acceleration, $a$, at a displacement, $y$, from $\text{O}$ satisfies the differential equation

$a = -\omega^2 y, \text{ where } \omega > 0.$

The buoy's displacement, $y$, from $\text{O}$ at time $t$ is given by

$y = H\sin(\omega t + c), \text{ where } t \ge 0,\ H, \omega > 0 \text{ and } -\pi \le c \le \pi.$

By finding expressions for $\frac{\mathrm{d}y}{\mathrm{d}t}$ and $\frac{\mathrm{d}^2y}{\mathrm{d}t^2}$, verify that $y = H\sin(\omega t + c)$ satisfies the differential equation $a = -\omega^2 y$.

Use the chain rule to show that $a = v\frac{\mathrm{d}v}{\mathrm{d}y}$, where $v$ is velocity.

By solving the differential equation, $v\frac{\mathrm{d}v}{\mathrm{d}y} = -\omega^2 y$, show that $v^2 = \omega^2(H^2 - y^2)$.

Hence, or otherwise, find the buoy's maximum speed.

The continuous random variable $Y$ denotes the buoy's displacement, $y$, from $\text{O}$ at time $t$.

The probability density function $f$ of $Y$ is defined by

$f(y) = \begin{cases} \frac{1}{\pi\sqrt{H^2 - y^2}}, & -H < y < H \\ 0, & \text{otherwise.} \end{cases}$

Show that $\mathrm{P}\left(0 \le Y \le \frac{H\sqrt{3}}{2}\right) = \frac{1}{3}$.

For $-H < y < H$, the function $f(y)$ can be expressed in the form $\frac{m}{|v(y)|}$, where $m > 0$ and $v(y)$ is the buoy's velocity at a displacement, $y$, from $\text{O}$.

Find the value of $m$.

Determine $\mathrm{E}(Y)$, justifying your answer.

Interpret the result found in part (f)(i) in the context of the buoy's motion.

If a company is producing cylindrical cans to package a new drink such that the volume $V$ of the can is $1000$ $cm^{3}$ and the cost of the can is determined by the surface area $S$ controlled by the radius $r$ and height $h$:

$a$ Find an expression for the surface area of the can in terms of the radius only $S(r)$.

$b$ Find the dimensions that would minimize the surface area.

$c$ Find the minimum surface area.

If a company is designing a triangular picnic area bordered by a stone wall on two sides, with the third side along a river that needs no wall. Let $a$ and $b$ represent the lengths of the two sides of the triangle that require a wall.

$a$ If the company has 80 meters of wall material available, find an equation for its area $A(a)$.

$b$ Find the dimensions that would maximize the area.

$c$ Find maximum area size.

If a landscaper is designing a rectangular flower bed that will be surrounded by a path of uniform width of 2 meters only from the length sides of the rectangle with a total outer perimeter of 120 meters. Let $x$ and $y$ be the width and length of the flower bed respectively.

$a$ Find an expression for the proposed flower bed design in terms of its width as $A(x)$.

$b$ Find the maximum area.