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Q17:
The minute hand of a circular clock is $15$ cm long. How far does the tip of the minute hand move in $1$ hour. (Take $\pi = 3.14$)

Solution :

Given Variables & Initial Setup

To determine the distance covered by the tip of the minute hand, we must first establish the geometric parameters of the system based on the provided values:

  • Length of the minute hand (Radius, $r$) = $15$ cm
  • Time elapsed ($t$) = $1$ hour
  • Value of Pi ($\pi$) = $3.14$

Step 1: Analyzing the Kinematics of the Minute Hand

The minute hand of a clock is anchored at the center of the clock face. As time progresses, the tip of the minute hand traces a perfect circle. [Per the definition of a circle: the locus of all points in a plane that are at a given distance from a given point, the center].

In exactly $1$ hour ($60$ minutes), the minute hand completes one full revolution around the clock face. Therefore, the angular displacement is $360^\circ$ or $2\pi$ radians. The total linear distance moved by the tip of the minute hand is exactly equal to the circumference of the circle it traces.

Step 2: Formulating the Geometric Equation

The circumference ($C$) of a circle is given by the standard geometric formula:

$C = 2\pi r$

Where:

  • $C$ represents the total distance moved by the tip in one revolution.
  • $\pi$ is the mathematical constant (given as $3.14$).
  • $r$ is the radius of the circular path, which corresponds to the length of the minute hand ($15$ cm).

Step 3: Visualizing the Circular Path

The following diagram illustrates the circular path traced by the tip of the minute hand over the course of $1$ hour.

r = 15 cm Distance = 2Ï€r (1 Revolution) 12 3 6 9

Step 4: Execution and Algebraic Calculation

Substitute the given values into the circumference formula:

$C = 2 \times 3.14 \times 15$

To simplify the calculation, use the commutative property of multiplication to group the integers first:

$C = (2 \times 15) \times 3.14$

$C = 30 \times 3.14$

Perform the final multiplication:

$C = 94.2$ cm

Final Solution: The tip of the minute hand moves a total distance of $94.2$ cm in $1$ hour.


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