A lighthouse L is located offshore, 500 metres from the nearest point P on a long straight shoreline. The narrow beam of light from the lighthouse rotates at a constant rate of $8\pi$ radians per minute, producing an illuminated spot S that moves along the shoreline. You may assume that the height of the lighthouse can be ignored and that the beam of light lies in the horizontal plane defined by sea level.

When S is 2000 metres from P,

(a)     show that the speed of S, correct to three significant figures, is ${\text{214}}\,{\text{000}}$ metres per minute;

(b)     find the acceleration of S.

(a)     the distance of the spot from P is $x = 500\tan \theta$     A1

the speed of the spot is

$\frac{{{\text{d}}x}}{{{\text{d}}t}} = 500{\sec ^2}\theta \frac{{{\text{d}}\theta }}{{{\text{d}}t}}{\text{ }}( = 4000\pi {\sec ^2}\theta )$     M1A1

when $x = 2000,{\text{ }}{\sec ^2}\theta = 17{\text{ }}(\theta = 1.32581 \ldots )\left( {\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 8\pi } \right)$

$\Rightarrow \frac{{{\text{d}}x}}{{{\text{d}}t}} = 500 \times 17 \times 8\pi$     M1A1

speed is ${\text{214 000}}$ (metres per minute)     AG

Note: If their displayed answer does not round to ${\text{214 000}}$, they lose the final A1.

(b)     $\frac{{{{\text{d}}^2}x}}{{{\text{d}}{t^2}}} = 8000\pi {\sec ^2}\theta \tan \theta \frac{{{\text{d}}\theta }}{{{\text{d}}t}}$ or $500 \times 2{\sec ^2}\theta \tan \theta {\left( {\frac{{{\text{d}}\theta }}{{{\text{d}}t}}} \right)^2}$     M1A1

$$\left( {{\text{since }}\frac{{{{\text{d}}^2}\theta }}{{{\text{d}}{t^2}}} = 0} \right)$$

$= {\text{43}}\,{\text{000}}\,{\text{000 (}} = 4.30 \times {10^7}){\text{ (metres per minut}}{{\text{e}}^2})$     A1

[8 marks]

This was a wordy question with a clear diagram, requiring candidates to state variables and do some calculus. Very few responded appropriately.