Mathematics 10 — Exercise Review · Q7

Full question

A pilot flies 200 km at bearing 045°, then 150 km at bearing 135°. Find distance from the start.

Step-by-step solution

1. Resolve each bearing leg: east=d sinβ, north=d cosβ. Add east components and north components independently.

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

   $$(247.487373415292,35.3553390593274)$$

3. Calculate this operation before combining the remaining terms.

   $${\left(\frac{61871843353823}{250000000000}\right)}^2=\frac{3828125000000011336768715329}{62500000000000000000000}$$

4. Calculate this operation before combining the remaining terms.

   $${\left(\frac{176776695296637}{5000000000000}\right)}^2=\frac{31250000000000042037423509769}{25000000000000000000000000}$$

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

   $$\mathrm{LCD}=25000000000000000000000000$$

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

   $$\frac{3828125000000011336768715329}{62500000000000000000000}=\frac{3828125000000011336768715329\times 400}{62500000000000000000000\times 400}$$

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

   $$\frac{1531250000000004534707486131600}{25000000000000000000000000}$$

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

   $$\frac{31250000000000042037423509769}{25000000000000000000000000}=\frac{31250000000000042037423509769\times 1}{25000000000000000000000000\times 1}$$

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

   $$\frac{31250000000000042037423509769}{25000000000000000000000000}$$

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

    $$\frac{1531250000000004534707486131600+31250000000000042037423509769}{25000000000000000000000000}=\frac{1562500000000004576744909641369}{25000000000000000000000000}$$

11. Calculate this operation before combining the remaining terms.

    $$\frac{3828125000000011336768715329}{62500000000000000000000}+\frac{31250000000000042037423509769}{25000000000000000000000000}=\frac{1562500000000004576744909641369}{25000000000000000000000000}$$

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

    $${247.487373415292}^2+{35.3553390593274}^2$$

13. Take the positive square root.

    $$250.0$$

14. Calculate this operation before combining the remaining terms.

    $$\frac{1}{\frac{35355339}{1000000}}=\frac{1000000}{35355339}$$

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

    $$\frac{24748737\times 1000000}{100000\times 35355339}$$

16. Calculate these two products separately.

    $$\frac{24748737000000}{3535533900000}$$

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

    $$\mathrm{GCD}=300000$$

18. Show the two divisions explicitly.

    $$\frac{\frac{24748737000000}{300000}}{\frac{3535533900000}{300000}}$$

19. Evaluate the numerator division.

    $$\frac{24748737000000}{300000}=82495790$$

20. Evaluate the denominator division.

    $$\frac{3535533900000}{300000}=11785113$$

21. Write the reduced fraction.

    $$\frac{82495790}{11785113}$$

22. Calculate this operation before combining the remaining terms.

    $$\frac{24748737}{100000}\times \frac{1000000}{35355339}=\frac{82495790}{11785113}$$

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

    $$\frac{247.48737}{35.355339}$$

24. Use inverse tangent in degrees.

    $$81.869898$$

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

    $$\mathrm{bearing}=\mathrm{081.87°}$$

Answer

The final answer is shown below.