Question: A camel travelled across a desert at a bearing of $104^\circ$ from an oasis.
If the camel stopped $23$ km east of the oasis, how far south of the oasis did it stop?
Give your answer to $1$ d.p.
1. **State the problem:** A camel travels from an oasis at a bearing of $104^\circ$. It stops $23$ km east of the oasis. We need to find how far south it stopped.
2. **Understanding bearing:** Bearing is measured clockwise from north. A bearing of $104^\circ$ means the camel's path is $104^\circ$ clockwise from north.
3. **Set up the coordinate system:**
- East corresponds to the $x$-axis.
- North corresponds to the $y$-axis.
4. **Decompose the camel's travel into components:**
- The camel's position vector makes an angle of $104^\circ$ from north.
- The east (x) component is given by $d \sin(\theta)$.
- The south (y) component is given by $d \cos(\theta)$ but since south is negative $y$, we consider the vertical component as negative.
5. **Given:**
- East component $= 23$ km
- Bearing $\theta = 104^\circ$
6. **Find the distance traveled $d$ using the east component:**
$$ 23 = d \sin(104^\circ) $$
$$ d = \frac{23}{\sin(104^\circ)} $$
7. **Calculate $d$:**
$$ \sin(104^\circ) \approx 0.9703 $$
$$ d \approx \frac{23}{0.9703} \approx 23.7 \text{ km} $$
8. **Find the south component:**
$$ \text{south distance} = d \cos(104^\circ) $$
$$ \cos(104^\circ) \approx -0.2419 $$
$$ \text{south distance} = 23.7 \times (-0.2419) \approx -5.7 \text{ km} $$
9. **Interpretation:** The negative sign indicates south direction.
10. **Final answer:** The camel stopped approximately $5.7$ km south of the oasis.
**Answer:** $5.7$ km south of the oasis (to 1 d.p.)