11

11. [Maximum mark: 17]

A rotating sprinkler is at a fixed point $S$.
It waters all points inside and on a circle of radius 20 metres.
Point $S$ is 14 metres from the edge of a path which runs in a north‑south direction.
The edge of the path intersects the circle at points $A$ and $B$.
This information is shown in the following diagram (a circle centered at $S$ with radius 20 m, a vertical line representing the path 14 m from $S$, intersecting the circle at $A$ and $B$).

(a) Show that $AB = 28.57$, correct to four significant figures. [3]

The sprinkler rotates at a constant rate of one revolution every 16 seconds.

(b) Show that the sprinkler rotates through an angle of $\frac{\pi}{8}$ radians in one second. [1]

Let $T$ seconds be the time that $[AB]$ is watered in each revolution.

(c) Find the value of $T$. [4]

(Question 11 continued)
Consider one clockwise revolution of the sprinkler.
At $t = 0$, the water crosses the edge of the path at $A$.
At time $t$ seconds, the water crosses the edge of the path at a movable point $D$ which is a distance $d$ metres south of point $A$.
Let $\alpha = \widehat{A S D}$ and $\beta = \widehat{S A B}$, where $\alpha, \beta$ are measured in radians.
This information is shown in the following diagram (a diagram showing the sprinkler $S$, the path, points $A, B, D$, and angles $\alpha$ and $\beta$).

(d) Write down an expression for $\alpha$ in terms of $t$. [1]

It is known that $\beta = 0.7754$ radians, correct to four significant figures.

(e) By using the sine rule in $\Delta ASD$, show that the distance, $d$, at time $t$, can be modelled by $d(t) = \frac{20 \sin(\frac{\pi t}{8})}{\sin(2.37 - \frac{\pi t}{8})}$. [3]

(Question 11 continued)
A turtle walks south along the edge of the path.
At time $t$ seconds, the turtle’s distance, $g$ metres south of $A$, can be modelled by $g(t) = 0.05t^2 + 1.1t + 18$, where $t \geq 0$.

(f) At $t = 0$, state how far south the turtle is from $A$. [1]

Let $w$ represent the distance between the turtle and point $D$ at time $t$ seconds.

(g) (i) Use the expressions for $g(t)$ and $d(t)$ to write down an expression for $w$ in terms of $t$.

(ii) Hence find when and where on the path the water first reaches the turtle. [4]