Practice section 8 of 8

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]

Hence find exactly:

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

(iii) that maximum volume. [2]

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]

(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]

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

(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]