Problem 90 - Algebra and Trigonometry

Question

The minute hand of a clock is 6 inches long and moves from 12 to 4 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 travels approximately \(12.56\) inches.

Step 1: Understanding the movement of the minute hand

The minute hand of the clock moves from 12 to 4. This is one third of a full circle (since a full circle would involve 12 hours).

Step 2: Calculating the circumference of the full circle

The formula for the circumference of a circle is \(C= 2\pi r\), where \(r\) is the radius of the circle. Given that the length of the minute hand, which acts as the radius, is 6 inches, we find the full circumference as \(C= 2\pi * 6 = 12\pi\) inches.

Step 3: Calculating the distance the tip of the minute hand travels

Since 12 to 4 is one third of a full circle, the distance the tip of the minute hand travels is one third of the full circumference, which is \(12\pi /3 = 4\pi\) inches.

Step 4: Rounding

Finally, we express our answer in decimal form rounded to two decimal places. One can calculate the numerical value of \(\pi\) as being approximately 3.14. Multiply this by 4 to get \(4\pi = 12.56\) inches.