Question

An analog clock starts at midnight and runs for $2025$ minutes before stopping. What is the tangent of the acute angle between the hour hand and the minute hand when the clock stops?

$0$

$\sqrt{2} - 1$

$2 - \sqrt{2}$

$\dfrac{\sqrt{2}}{2}$

$3 - \sqrt{2}$

Answer: B

Concepts:clock trigonometric identity

Difficulty rating: 1840

Solution:

$2025$ minutes is $33$ hours $45$ minutes, which modulo $12$ hours reads $9{:}45.$ The minute hand points at $270^\circ$ and the hour hand at $9.75 \times 30^\circ = 292.5^\circ,$ so the acute angle between them is $22.5^\circ.$ Using the half-angle value, $\tan 22.5^\circ = \sqrt{2} - 1.$

Thus, the correct answer is B.

2025 AMC 12B Problem 16

Answer: B

Difficulty rating: 1840

Solution:

Problem 16 in Other Years