Exam-Style Questions on Optimisation

A child's play tent is made in the shape of half a cylinder. It is constructed from a fibreglass frame with material pulled tightly around it. The fibreglass frame consists of a rectangular base, two semi-circular ends and two further support rods, as shown in the following diagram.

The semicircular ends each have radius \(r\) and the support rods each have length \(d\).

Let F be the total length of fibreglass used in the frame of the play tent.

(a) Write down an expression for \(F\) in terms of \(r\), \(d\) and \(\pi\).

The volume of the play tent is 0.95 m³.

(b) Write down an equation for the volume of the play tent in terms of \(r\), \(d\) and \(\pi\).

(c) Show that \(F = 2\pi r + 4r + \frac{7.6}{\pi r^2}\)

(d) Find \(\frac{dF}{dr}\)

The play tent is designed so that the length of fibreglass used in its frame is a minimum.

(e) Find the value of \(r\) for which \(F\) is a minimum.

(f) Calculate the value of \(d\) for which \(F\) is a minimum.

(g) Calculate the minimum value of \(F\).

Worked Solution

A package is in the shape of a cuboid and has a length \(l\) cm, width \(w\) cm and height of 12 cm.

(a) Express the volume of the package in terms of \(l\) and \(w\).

The total volume of the package is 2400 cm³.

(b) Show that \(l=\frac{200}{w}\).

The package is tied up using a length of red string that fits exactly around the package in two different directions, as shown in the following diagram (not to scale).

(c) Show that the length of string, \(x\)cm, required to tie up the package can be written as \(24+4w+\frac{400}{w}\)

(d) Sketch the graph of \(x\) for \(0\lt w \le 12\), clearly showing the local minimum point.

(e) Find \(\frac{dx}{dw}\).

(f) Find the value of \(w\) for which \(x\) is a minimum.

(g) Find the value, \(l\), of the package for which the length of string is a minimum.

(h) Find the minimum length of string required to tie up the package.

Worked Solution

A circle with equation \(x^2+y^2=25 \) has centre \((0,0)\) and radius 5.

A trapezium, ABCD, is inscribed in the circle with its vertices at \( A(x,y), B(4,-3), C(-4,-3) \text{ and } D(-x,y)\), where A and D are variable points in the first and second quadrants respectively. This is shown in the following sketch.

(a) For point A, show that \(y = \sqrt{25-x^2} \).

(b) Hence, find an expression for T, the area of trapezium ABCD, in terms of x.

(c) Show that \( \dfrac{dT}{dx} = 3 - \dfrac{2x^2 +4x -25}{\sqrt{25-x^2}} \).

(d) Hence or otherwise, find the value of \(x\) such that T is a maximum.

Worked Solution

Moresum Soup is sold in cans with a capacities of 400ml each. Each can is in the shape of a right circular cylinder with radius \(r\) cm and height \(h\) cm.

(a) Assuming that the can is made from a metal of negligible thickness prove that the total surface area, A cm², of the can is given by the following formula:

(b) Given that r can vary, find the dimensions of a can that has minimum surface area.

Worked Solution

The north corridor at Addwell High School has a width of \(2 \, \text{m}\). There is a ninety-degree corner at point \(C\). Points \(A\) and \(B\) are variable points on the base of the walls such that \(A\), \(C\), and \(B\) lie on a straight line.

Let \(L\) denote the length \(AB\) in metres.

Let \(\theta\) be the angle that \([AB]\) makes with the corridor wall, where \(0 < \theta < \frac{\pi}{2}\).

(a) Find the length of L in terms of \(\theta\).

(b) Find \(\frac{dL}{d\theta}\).

(c) When \(\frac{dL}{d\theta} = 0\), show that \(\theta = \frac{\pi}{4} \).

(d) Find \(\frac{d^2 L}{d\theta^2}\).

(e) Find \(\frac{d^2 L}{d\theta^2}\) when \(\theta = \frac{\pi}{4} \).

(f) Hence, justify that \(L\) is a minimum when \(\theta = \frac{\pi}{4} \).

(g) Determine this minimum value of \(L\).

Two people need to carry a pipe of length \(7 \, \text{m}\) along this corridor. The height of the corridor is \(2.5 \, \text{m}\).

(h) Determine whether this is possible, giving a reason for your answer.

(i) If the width of the pipe is considered to be negligible, what is theoretically the maximum length of a pipe that is able to be carried around the corner of the corridor?

Worked Solution