1. **Problem 5:** At what time between 4 and 5 o'clock will the hands of the clock be perpendicular (i.e., angle $\theta=90^{\circ}$)? Provide both possible times. 2. Recall the formula for angle between the hands: $$\theta=30H - \frac{11}{2}M$$ where $H$ is the hour and $M$ is the minutes after the hour. 3. For between 4 and 5 o'clock, $H=4$, so: $$90 = 30 \times 4 - \frac{11}{2}M$$ $$90 = 120 - \frac{11}{2}M$$ 4. Solve for $M$: $$\frac{11}{2}M = 120 - 90 = 30$$ $$M = \frac{30 \times 2}{11} = \frac{60}{11} \approx 5.45 \text{ minutes}$$ 5. Since hands can be perpendicular on two positions in an hour, use also the other case: $$\theta = 180 - 90 = 90^{\circ}$$ (hands form perpendicular both ways) Using $\theta=90$ or $\theta=270$ (angle between can be $90^{\circ}$ or $270^{\circ}$ for perpendicular): For $\theta=270$: $$270 = 120 - \frac{11}{2}M$$ $$\frac{11}{2}M = 120 - 270 = -150$$ $$M = \frac{-150 \times 2}{11} = -\frac{300}{11} \approx -27.27 \text{ (not valid as minutes)}$$ Instead, use $\theta = |30H - \frac{11}{2}M| = 90$, so also: $$30H - \frac{11}{2}M = -90$$ $$120 - \frac{11}{2}M = -90$$ $$\frac{11}{2}M = 120 + 90 =210$$ $$M = \frac{210 \times 2}{11} = \frac{420}{11} \approx 38.18 \text{ minutes}$$ 6. Thus, the two times when the clocks are perpendicular between 4 and 5 are approximately 4:5.45 and 4:38.18. --- 7. **Problem 6:** Use Padi's formula to find the time between 9 and 10 o'clock when the minute hand is exactly 2 minute spaces ahead of the hour hand. 8. Padi's formula: $$M = \frac{2}{11} (\text{reference} \pm \text{required})$$ 9. Define terms: - reference = $30H = 30 \times 9 = 270^{\circ}$ (hour hand's angle at 9:00) - required = $2$ minute spaces (difference between minute and hour hand positions) 10. Since minute hand is ahead, use plus: $$M = \frac{2}{11} (270 + 2) = \frac{2}{11} \times 272 = \frac{544}{11} \approx 49.45 \text{ minutes}$$ 11. Therefore, time is approximately 9:49.45 (9:49 and 27 seconds). **Final answers:** - Problem 5: Times are approximately **4:5.45** and **4:38.18**. - Problem 6: Time is approximately **9:49.45**.