Review practice · Section 2 of 2

The line $y=mx+3/m$ is tangent to the parabola $y^2=kx$ for every $m\ne0$. Find $k$ and the coordinates of contact in terms of $m$.4 marks

Sand from a chute forms a conical pile whose altitude is always $4/3$ of its base radius.

(i) How fast is its volume increasing when the radius is $1$ m and increases at $1/8\text{ cm s}^{-1}$?4 marks

(ii) How fast is its radius increasing when it is $2$ m and the volume increases at $1000\text{ cm}^3\text{ s}^{-1}$?3 marks

Point $Q(x,y)$ lies on $4x^2+xy+y^2=36$, with $y\ge0$. The curve meets the x-axis at $P,R$ as drawn.

(i) Show that triangle $PQR$ has area $A=3y$.3 marks

(ii) Find $dy/dx$ in terms of $x,y$.2 marks

(iii) Hence find the x-value where $A$ is stationary and use the second derivative test to determine its nature.6 marks

If $x$ increases at $8$ units/s, find the rate of change of $A$ when $x=0$.3 marks

A flower-bed for a large garden consists of an $x$ m by $y$ m rectangle topped by a semicircle of diameter $x$ m. A low wall takes $3$ hours per metre along the straight parts and $9$ hours per metre around the semicircular arc. A total of $180$ hours is available.

Use differentiation to find $x,y$ giving maximum area.10 marks

A decorative monument for the 2010 Youth Olympic Games consists of a complete globe on a closed cylindrical base of radius $r$ and height $h$. The globe radius is $1.25r$. To minimise cost, the total surface area is fixed at $36.5\text{ m}^2$.

(i) Show that $h=(146-33\pi r^2)/(8\pi r)$.2 marks

Hence find exactly:

(ii) the cylinder radius giving greatest total volume.7 marks

(iii) that maximum volume.2 marks

A sphere of radius $R$ has area $4\pi R^2$ and volume $4\pi R^3/3$.

A square lawn $ABCD$ has side $26$ m and a circular pond of radius $11$ m centred at its centre $P$. A rectangular flower crate $QTCS$ has corner $C$ at the square corner and opposite corner $Q$ on the pond. Let $R$ be the midpoint of $DC$ on the downward centre line. The angle $RPQ=\theta$ is measured from $PR$, with $0\le\theta\le\pi/4$. Its base area is $X\text{ m}^2$.

(i) Express $X$ in terms of $\theta$ and show $dX/d\theta=11(\sin\theta-\cos\theta)(13-11\sin\theta-11\cos\theta)$.3 marks

(ii) For stationary values, show the angles satisfy $\tan\theta_1=1$ and $\sin(\theta_2+\alpha)=k$, where $k,\alpha$ are exact constants. Find $\theta_1,\theta_2$.4 marks

(iii) Classify both stationary values and find their areas.3 marks

(iv) Four identical crates are placed at $A,B,C,D$, with the rest outside the pond covered by grass. Find the smallest grass area to three significant figures.3 marks