Find the linear speed \( v \) of the tip of the minute hand of a clock, if the hand is 9 cm long. \( v=\square \mathrm{cm} \) per minute (Simplify your answer. Type an exact answer, using \( \pi \) as needed. Use integers or fractions for any numbers in the

To find the linear speed \( v \) of the tip of the minute hand of a clock, we can use the formula for linear speed, which is given by: \[ v = \frac{d}{t} \] where \( d \) is the distance traveled and \( t \) is the time taken. 1. **Calculate the distance traveled by the tip of the minute hand in one complete revolution:** The minute hand makes one complete revolution in 60 minutes. The distance traveled in one complete revolution is the circumference of the circle traced by the tip of the minute hand. The formula for the circumference \( C \) of a circle is: \[ C = 2\pi r \] where \( r \) is the radius of the circle. In this case, the length of the minute hand is the radius, which is 9 cm. Therefore, the circumference is: \[ C = 2\pi \times 9 = 18\pi \text{ cm} \] 2. **Determine the time taken for one complete revolution:** The minute hand takes 60 minutes to complete one revolution. 3. **Calculate the linear speed \( v \):** Now we can substitute the values into the formula for linear speed: \[ v = \frac{d}{t} = \frac{18\pi \text{ cm}}{60 \text{ min}} = \frac{18\pi}{60} \text{ cm/min} \] Simplifying this fraction: \[ v = \frac{3\pi}{10} \text{ cm/min} \] Thus, the linear speed \( v \) of the tip of the minute hand is: \[ \boxed{\frac{3\pi}{10}} \text{ cm/min} \]