(34) A 6 FOOT (IN DIAMETER) ROTATES AT HOO REVOLUTIONS PER MIMUTE (a) FIND THE ANGULAR SPEED IN RADIANS PER MINUTE (b) FIND LINEAR SPEED IN FEET PER MINUTE

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To solve the problem, we need to find both the angular speed in radians per minute and the linear speed in feet per minute for a circle with a diameter of 6 feet that rotates at 100 revolutions per minute (RPM).

### Step 1: Extract Known Conditions
- Diameter of the circle, $$ d = 6 $$ feet
- Radius of the circle, $$ r = \frac{d}{2} = \frac{6}{2} = 3 $$ feet
- Revolutions per minute, $$ n = 100 $$ RPM

### Step 2: Find Angular Speed in Radians per Minute
Angular speed $$ \omega $$ in radians per minute can be calculated using the formula:
$$
\omega = n \times 2\pi
$$
where $$ n $$ is the number of revolutions per minute and $$ 2\pi $$ is the number of radians in one revolution.

### Step 3: Find Linear Speed in Feet per Minute
Linear speed $$ v $$ can be calculated using the formula:
$$
v = r \times \omega
$$
where $$ r $$ is the radius and $$ \omega $$ is the angular speed in radians per minute.

Now, let's perform the calculations.

1. Calculate the angular speed $$ \omega $$.
2. Calculate the linear speed $$ v $$.

Let's compute these values.
Calculate the value by following steps:
- step0: Calculate:
  $$3\left(100\times 2\pi \right)$$
- step1: Remove the parentheses:
  $$3\times 100\times 2\pi $$
- step2: Multiply the terms:
  $$600\pi $$
Calculate or simplify the expression $$ 100 * 2 * \pi $$.
Calculate the value by following steps:
- step0: Calculate:
  $$100\times 2\pi $$
- step1: Multiply the terms:
  $$200\pi $$
### Step 4: Results

1. **Angular Speed**:
   - The angular speed $$ \omega $$ in radians per minute is:
   $$
   \omega = 100 \times 2\pi \approx 628.32 \text{ radians per minute}
   $$

2. **Linear Speed**:
   - The linear speed $$ v $$ in feet per minute is:
   $$
   v = 3 \times (100 \times 2\pi) \approx 1884.96 \text{ feet per minute}
   $$

### Final Answers:
- (a) The angular speed is approximately $$ 628.32 $$ radians per minute.
- (b) The linear speed is approximately $$ 1884.96 $$ feet per minute.
The angular speed is approximately 628.32 radians per minute, and the linear speed is approximately 1884.96 feet per minute.

(34) A 6 FOOT (IN DIAMETER) ROTATES AT HOO REVOLUTIONS PER MIMUTE (a) FIND THE ANGULAR SPEED IN RADIANS PER MINUTE (b) FIND LINEAR SPEED IN FEET PER MINUTE

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