Mathematics 10 — Exercise 7.4 · Q7

Full question

Fatima walks 5.5 km south then 1.3 km west. Find her bearing from the start (the book changes her name to Huria in the last sentence).

Step-by-step solution

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

   $$(-1.3,-5.5)$$

2. Calculate this operation before combining the remaining terms.

   $${\left(-\frac{13}{10}\right)}^2=\frac{169}{100}$$

3. Calculate this operation before combining the remaining terms.

   $${\left(-\frac{11}{2}\right)}^2=\frac{121}{4}$$

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

   $$\mathrm{LCD}=100$$

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

   $$\frac{169}{100}=\frac{169\times 1}{100\times 1}$$

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

   $$\frac{169}{100}$$

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

   $$\frac{121}{4}=\frac{121\times 25}{4\times 25}$$

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

   $$\frac{3025}{100}$$

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

   $$\frac{169+3025}{100}=\frac{3194}{100}$$

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

    $$\mathrm{GCD}=2$$

11. Show the two divisions explicitly.

    $$\frac{\frac{3194}{2}}{\frac{100}{2}}$$

12. Evaluate the numerator division.

    $$\frac{3194}{2}=1597$$

13. Evaluate the denominator division.

    $$\frac{100}{2}=50$$

14. Write the reduced fraction.

    $$\frac{1597}{50}$$

15. Calculate this operation before combining the remaining terms.

    $$\frac{169}{100}+\frac{121}{4}=\frac{1597}{50}$$

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

    $${\left(-1.3\right)}^2+{\left(-5.5\right)}^2$$

17. Take the positive square root.

    $$5.6515485$$

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

    $$\frac{1.3}{5.5}$$

19. Use inverse tangent in degrees.

    $$13.29857$$

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

    $$\mathrm{bearing}=\mathrm{193.30°}$$

Answer

The final answer is shown below.