Mathematics 10 — Exercise 7.4 · Q9

Full question

A ship travels 100 km at bearing 045°, then 120 km at bearing 135°. Find its distance from the start.

Step-by-step solution

1. Bearing components are east=d sinβ and north=d cosβ. Calculate each leg separately, then add matching components.

2. Take east as the horizontal component and north as the vertical component.

   $$(155.56349186104,-14.1421356237309)$$

3. Calculate this operation before combining the remaining terms.

   $${\left(\frac{1944543648263}{12500000000}\right)}^2=\frac{3781249999999977862917169}{156250000000000000000}$$

4. Calculate this operation before combining the remaining terms.

   $${\left(-\frac{141421356237309}{10000000000000}\right)}^2=\frac{19999999999999857198323561481}{100000000000000000000000000}$$

5. To add these fractions, first use a common denominator: the least common multiple of their denominators.

   $$\mathrm{LCD}=100000000000000000000000000$$

6. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

   $$\frac{3781249999999977862917169}{156250000000000000000}=\frac{3781249999999977862917169\times 640000}{156250000000000000000\times 640000}$$

7. Calculate the converted numerator; the denominator is now common.

   $$\frac{2419999999999985832266988160000}{100000000000000000000000000}$$

8. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

   $$\frac{19999999999999857198323561481}{100000000000000000000000000}=\frac{19999999999999857198323561481\times 1}{100000000000000000000000000\times 1}$$

9. Calculate the converted numerator; the denominator is now common.

   $$\frac{19999999999999857198323561481}{100000000000000000000000000}$$

10. Keep the common denominator and add the signed numerators.

    $$\frac{2419999999999985832266988160000+19999999999999857198323561481}{100000000000000000000000000}=\frac{2439999999999985689465311721481}{100000000000000000000000000}$$

11. Calculate this operation before combining the remaining terms.

    $$\frac{3781249999999977862917169}{156250000000000000000}+\frac{19999999999999857198323561481}{100000000000000000000000000}=\frac{2439999999999985689465311721481}{100000000000000000000000000}$$

12. Square both components and add for the squared displacement.

    $${155.56349186104}^2+{\left(-14.1421356237309\right)}^2$$

13. Take the positive square root.

    $$156.20499$$

14. Calculate this operation before combining the remaining terms.

    $$\frac{1}{\frac{1767767}{125000}}=\frac{125000}{1767767}$$

15. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{15556349\times 125000}{100000\times 1767767}$$

16. Calculate these two products separately.

    $$\frac{1944543625000}{176776700000}$$

17. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=25000$$

18. Show the two divisions explicitly.

    $$\frac{\frac{1944543625000}{25000}}{\frac{176776700000}{25000}}$$

19. Evaluate the numerator division.

    $$\frac{1944543625000}{25000}=77781745$$

20. Evaluate the denominator division.

    $$\frac{176776700000}{25000}=7071068$$

21. Write the reduced fraction.

    $$\frac{77781745}{7071068}$$

22. Calculate this operation before combining the remaining terms.

    $$\frac{15556349}{100000}\times \frac{125000}{1767767}=\frac{77781745}{7071068}$$

23. Find the reference angle from north/south using tan θ=|east|/|north|.

    $$\frac{155.56349}{14.142136}$$

24. Use inverse tangent in degrees.

    $$84.805571$$

25. Place the reference angle in its quadrant and measure clockwise from NORTH.

    $$\mathrm{bearing}=\mathrm{095.19°}$$

Answer

The final answer is shown below.