Final answer:
The ratio of the distances traveled by the tips of the hour hand to the minute hand from noon to 3 p.m. is 3/16. This was determined by calculating the arc lengths for each hand during this time period using the formulas for arc length and circumference of a circle.
Step-by-step explanation:
To find the ratio of the distances traveled by the tips of the hour hand and the minute hand from noon to 3 p.m., we first need to calculate the lengths of the arcs that each hand has swept during this time. Since the clock makes a full rotation every 12 hours and the hour hand has moved a quarter of this distance by 3 p.m., it has traveled through an angle of 90 degrees (1/4 of 360 degrees).
The distance traveled by the tip of the hour hand can be calculated using the formula for the arc length of a circle, which is arc length = radius × angle in radians. First, we convert 90 degrees to radians by the formula radians = degrees × (\pi / 180), which gives us \pi / 2 radians for 90 degrees. Then we multiply the length of the hour hand (6 inches) by the angle in radians to get the arc length: 6 inches × (\pi / 2) inches.
The minute hand rotates through 360 degrees every 60 minutes, so from noon to 3 p.m., it completes a full rotation. Therefore, the arc length for the minute hand is simply the circumference of the circle it describes, which can be calculated with the formula circumference = 2 × \pi × radius. The circumference for the minute hand is 2 × \pi × 8 inches.
Now, let's calculate the arc lengths: For the hour hand, we get 6 × (\pi / 2) = 3\pi inches, and for the minute hand, we get 2 × \pi × 8 = 16\pi inches. The ratio of the distances is the arc length of the hour hand divided by the arc length of the minute hand: 3\pi / 16\pi, which simplifies to 3/16. So, the ratio of the distances traveled by the tips of the hour hand to the minute hand from noon to 3 p.m. is 3/16.