A 78-rpm record turns at the rate of 78 revolutions per minute, as its name implies. Through what angle (in radians) would such a record turn during a song that lasted for 3 minutes and 15 seconds?
The Correct Answer and Explanation is:
To solve this problem, we need to determine how many radians the record turns during the duration of the song. Here’s how we approach it:
Step 1: Understand the given data
- The record turns at 78 revolutions per minute (rpm).
- The duration of the song is 3 minutes and 15 seconds. First, we need to convert this into a uniform unit of time, which is minutes.
3 minutes+15 seconds=3 minutes+1560 minutes=3.25 minutes3 \text{ minutes} + 15 \text{ seconds} = 3 \text{ minutes} + \frac{15}{60} \text{ minutes} = 3.25 \text{ minutes}3 minutes+15 seconds=3 minutes+6015 minutes=3.25 minutes
Step 2: Calculate the number of revolutions in 3.25 minutes
Since the record turns at 78 rpm, the number of revolutions in 3.25 minutes is:Number of revolutions=78×3.25=253.5 revolutions\text{Number of revolutions} = 78 \times 3.25 = 253.5 \text{ revolutions}Number of revolutions=78×3.25=253.5 revolutions
Step 3: Convert revolutions to radians
One full revolution corresponds to an angle of 2π2\pi2π radians. Therefore, to find the total angle in radians, we multiply the number of revolutions by 2π2\pi2π:Angle in radians=253.5×2π=253.5×6.2832≈1597.23 radians\text{Angle in radians} = 253.5 \times 2\pi = 253.5 \times 6.2832 \approx 1597.23 \text{ radians}Angle in radians=253.5×2π=253.5×6.2832≈1597.23 radians
Final Answer:
The record turns through approximately 1597.23 radians during the song that lasts 3 minutes and 15 seconds.
