Problem 100 - Algebra and Trigonometry

Question

On a carousel, the outer row of animals is 20 feet from the center. The inner row of animals is 10 feet from the center. The carousel is rotating at 2.5 revolutions per minute. What is the difference, in feet per minute, in the linear speeds of the animals in the outer and inner rows? Round to the nearest foot per minute.

Step-by-Step Solution

Answer

To get the solution, perform the calculations in each step. The difference in linear speeds of the animals in the outer and inner rows, when rounded to the nearest foot per minute, gives the final answer.

Step 1: Calculate the angular speed in radians per minute

The carousel is rotating at 2.5 revolutions per minute. To convert this to radians per minute, multiply by \( 2\pi \) because one revolution is \( 2\pi \) radians. Hence, the angular speed \( \omega \) in radians per minute is \( 2.5 \times 2\pi \) rad/min.

Step 2: Calculate the linear speeds for the outer and inner circles

Using the formula \( v = \omega r \), first calculate the linear speed of the animals in the outer row by substituting \( \omega \) from step 1 and \( r = 20 \) feet. Next, calculate the linear speed of the animals in the inner row by substituting \( \omega \) from step 1 and \( r = 10 \) feet.

Step 3: Calculate the difference in linear speeds

Subtract the linear speed of the animals in the inner row from that of the outer row to get the difference.

Step 4: Round to the nearest foot per minute

Finally, round off the difference obtained in step 3 to the nearest foot per minute as per the problem's instructions.