Mathematics 10 — Exercise 7.4 · Q8

Full question

An aircraft flies 150 km east then 100 km northeast, 45° north of east. Find the total displacement.

Step-by-step solution

1. Resolve the northeast leg into east=100cos45° and north=100sin45°.

2. Add the initial 150 km east to the second east component.

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

   $$(220.710678118655,70.7106781186548)$$

4. Calculate this operation before combining the remaining terms.

   $${\left(\frac{44142135623731}{200000000000}\right)}^2=\frac{1948528137423861400410360361}{40000000000000000000000}$$

5. Calculate this operation before combining the remaining terms.

   $${\left(\frac{176776695296637}{2500000000000}\right)}^2=\frac{31250000000000042037423509769}{6250000000000000000000000}$$

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

   $$\mathrm{LCD}=25000000000000000000000000$$

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

   $$\frac{1948528137423861400410360361}{40000000000000000000000}=\frac{1948528137423861400410360361\times 625}{40000000000000000000000\times 625}$$

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

   $$\frac{1217830085889913375256475225625}{25000000000000000000000000}$$

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

   $$\frac{31250000000000042037423509769}{6250000000000000000000000}=\frac{31250000000000042037423509769\times 4}{6250000000000000000000000\times 4}$$

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

    $$\frac{125000000000000168149694039076}{25000000000000000000000000}$$

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

    $$\frac{1217830085889913375256475225625+125000000000000168149694039076}{25000000000000000000000000}=\frac{1342830085889913543406169264701}{25000000000000000000000000}$$

12. Calculate this operation before combining the remaining terms.

    $$\frac{1948528137423861400410360361}{40000000000000000000000}+\frac{31250000000000042037423509769}{6250000000000000000000000}=\frac{1342830085889913543406169264701}{25000000000000000000000000}$$

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

    $${220.710678118655}^2+{70.7106781186548}^2$$

14. Take the positive square root.

    $$231.76109$$

15. Calculate this operation before combining the remaining terms.

    $$\frac{1}{\frac{35355339}{500000}}=\frac{500000}{35355339}$$

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

    $$\frac{5517767\times 500000}{25000\times 35355339}$$

17. Calculate these two products separately.

    $$\frac{2758883500000}{883883475000}$$

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

    $$\mathrm{GCD}=25000$$

19. Show the two divisions explicitly.

    $$\frac{\frac{2758883500000}{25000}}{\frac{883883475000}{25000}}$$

20. Evaluate the numerator division.

    $$\frac{2758883500000}{25000}=110355340$$

21. Evaluate the denominator division.

    $$\frac{883883475000}{25000}=35355339$$

22. Write the reduced fraction.

    $$\frac{110355340}{35355339}$$

23. Calculate this operation before combining the remaining terms.

    $$\frac{5517767}{25000}\times \frac{500000}{35355339}=\frac{110355340}{35355339}$$

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

    $$\frac{220.71068}{70.710678}$$

25. Use inverse tangent in degrees.

    $$72.235724$$

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

    $$\mathrm{bearing}=\mathrm{072.24°}$$

Answer

The final answer is shown below.