At 8:30, the hour hand and the minute hand of clock form an angle of

### Concept & Clock Hand Angles The angle $\theta$ formed between the hands of a clock at $H$ hours and $M$ minutes is calculated using the relative speeds of the minute ($6^\circ$/min) and hour ($0.5^\circ$/min) hands. $$ \theta = |30H - 5.5M| $$ ### Step-by-Step Solution * Given: Hours ($H$) = 8, Minutes ($M$) = 30. * Substitute the values into the angle formula: $\theta = |30(8) - 5.5(30)|$ * Evaluate the individual parts: $30 \times 8 = 240$ $5.5 \times 30 = 165$ * Subtract to find the absolute angle: $\theta = |240 - 165| = 75^\circ$. ### Exam Strategy & Shortcut Use the visualization method: At 8:30, the minute hand points exactly at 6 ($180^\circ$ from 12). The hour hand is halfway between 8 and 9. The angle from 6 to 8 is two 5-minute spaces ($2 \times 30^\circ = 60^\circ$). The hour hand has moved exactly half an hour ($30 \times 0.5^\circ = 15^\circ$) past the 8. Total angle = $60^\circ + 15^\circ = 75^\circ$. ### Common Pitfall Assuming the hour hand is exactly on the 8, which would incorrectly give an angle of $60^\circ$ (the angle between 6 and 8). ### Final Answer Therefore, the correct answer is **$75^\circ$**.