Problem 89 - Algebra and Trigonometry

Question

The minute hand of a clock is 8 inches long and moves from 12 to 2 o'clock. How far does the tip of the minute hand move? Express your answer in terms of \(\pi\) and then round to two decimal places.

Step-by-Step Solution

Answer

The tip of the minute hand moves approximately 8.38 inches from 12 to 2 o'clock.

Step 1: Find the full circumference

The full circumference \(C\) of a circle with radius \(r\) is given by the formula \(C = 2\pi r\). Since the radius of the circular path that the clock hand moves along is 8 inches, the full circumference is \(C = 2\pi * 8\).

Step 2: Calculate one-sixth of the circumference

Since the minute hand moves from 12 to 2, which is one-sixth of a full circle, therefore, we calculate a sixth of the full circumference: \(C_{16} = \frac{C}{6}\).

Step 3: Substitute the values

Substitute the value of \(C\) from step 1 into the equation: \(C_{16} = \frac{2\pi * 8}{6}\). Simplify the equation.

Step 4: Convert to decimal

Convert the result from step 3 to decimal form. Consider \(\pi\) to be approximately 3.14 to find the decimal equivalent.