Mathematics 10 — Exercise 9.2 · Q7

Full question

A Ferris wheel has radius 12 m and arc length 18 m. Find the central angle in radians.

Step-by-step solution

1. For radians, arc length l=rθ. Divide both sides by r to find θ=l/r.

2. Substitute the given numbers into the formula.

   $$\frac{18}{12}$$

3. Calculate this operation before combining the remaining terms.

   $$18\times 1=18$$

4. Calculate this operation before combining the remaining terms.

   $$1\times 12=12$$

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

   $$\frac{18\times 1}{1\times 12}$$

6. Calculate these two products separately.

   $$\frac{18}{12}$$

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

   $$\mathrm{GCD}=6$$

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

   $$\frac{18\times 1}{1\times 6}$$

9. Calculate these two products separately.

   $$\frac{18}{6}$$

10. Show the two divisions explicitly.

    $$\frac{\frac{18}{6}}{\frac{6}{6}}$$

11. Evaluate the denominator division.

    $$\frac{6}{6}=1$$

12. Write the reduced fraction.

    $$3\times \frac{1}{1}$$

13. Calculate this operation before combining the remaining terms.

    $$18\times \frac{1}{6}=3$$

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

    $$\frac{12\times 1}{1\times 6}$$

15. Calculate these two products separately.

    $$\frac{12}{6}$$

16. Show the two divisions explicitly.

    $$\frac{\frac{12}{6}}{\frac{6}{6}}$$

17. Write the reduced fraction.

    $$2\times \frac{1}{1}$$

18. Calculate this operation before combining the remaining terms.

    $$12\times \frac{1}{6}=2$$

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

    $$\frac{3\times 1}{1\times 2}$$

20. Calculate this operation before combining the remaining terms.

    $$3\times \frac{1}{2}=\frac{3}{2}$$

21. Show the two divisions explicitly.

    $$\frac{\frac{18}{6}}{\frac{12}{6}}$$

22. Evaluate the numerator division.

    $$\frac{18}{6}=3$$

23. Evaluate the denominator division.

    $$\frac{12}{6}=2$$

24. Write the reduced fraction.

    $$\frac{3}{2}$$

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

    $$\frac{1\times 1}{1\times 12}$$

26. Calculate these two products separately.

    $$\frac{1}{12}$$

27. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{12}=\frac{1}{12}$$

28. Calculate this operation before combining the remaining terms.

    $$18\times \frac{1}{12}=\frac{3}{2}$$

29. The required result is shown below. Unit: radians

    $$\frac{3}{2}$$

Answer

The final answer is shown below.