1. **Stating the problem:** We are given the formula for the speed of a tsunami: $$s = \sqrt{g \times d}$$ where $s$ is speed in metres per second, $d$ is ocean depth in metres, and $g = 9.8$ m/s². ### (a)(i) Find $s$ when $d = 2000$ m. 2. **Using the formula:** $$s = \sqrt{g \times d}$$ Substitute $g = 9.8$ and $d = 2000$: $$s = \sqrt{9.8 \times 2000}$$ 3. **Calculate inside the square root:** $$9.8 \times 2000 = 19600$$ 4. **Find the square root:** $$s = \sqrt{19600} = 140$$ 5. **Answer:** The speed of the tsunami is $140$ metres per second. --- ### (a)(ii) Find the time taken for the tsunami to travel 400 km to land. 6. **Convert distance to metres:** $$400 \text{ km} = 400 \times 1000 = 400000 \text{ m}$$ 7. **Use the formula for time:** $$\text{time} = \frac{\text{distance}}{\text{speed}} = \frac{D}{s}$$ 8. **Substitute values:** $$\text{time} = \frac{400000}{140}$$ 9. **Simplify the fraction:** $$\text{time} = \frac{\cancel{400000}}{\cancel{140}} = 2857.142857... \text{ seconds}$$ 10. **Convert seconds to minutes:** $$\frac{2857.142857}{60} \approx 47.62 \text{ minutes}$$ 11. **Answer:** Time taken is approximately $48$ minutes. --- ### (b)(i) Rearrange formula to express $d$ in terms of $g$ and $s$. 12. **Start with:** $$s = \sqrt{g \times d}$$ 13. **Square both sides:** $$s^2 = g \times d$$ 14. **Solve for $d$:** $$d = \frac{s^2}{g}$$ --- ### (b)(ii) Find $d$ when $s = 55$ m/s. 15. **Use rearranged formula:** $$d = \frac{s^2}{g} = \frac{55^2}{9.8}$$ 16. **Calculate numerator:** $$55^2 = 3025$$ 17. **Calculate depth:** $$d = \frac{3025}{9.8} \approx 308.67$$ 18. **Answer:** Depth is approximately $309$ metres to the nearest metre.