A clock has a 75 cm long second hand and a 60 cm long minute hand, respectively. In 30 minutes duration, the tip of the second hand will travel \(x\) distance more than the tip of the minute hand. The value of \(x\) in meters is nearly (Take \(\pi = 3.14\)):
- 139.4
- 140.5
- 220.0
- 118.9
The Correct Option is A
Approach Solution - 1
To find the additional distance traveled by the tip of the second hand compared to the minute hand in 30 minutes, we need to calculate the respective circumferences traveled by each hand and then find their difference.
- First, calculate the circumference traced by each hand:
- The second hand has a length of 75 cm. The circumference \(C_s\) traveled by the tip of the second hand in one complete rotation is given by: \(C_s = 2 \pi \times 75\). Using \(\pi = 3.14\), we get: \(C_s = 2 \times 3.14 \times 75 = 471 \text{ cm}\).
- The minute hand has a length of 60 cm. The circumference \(C_m\) traveled by the tip of the minute hand in one complete rotation is: \(C_m = 2 \pi \times 60\). Similarly: \(C_m = 2 \times 3.14 \times 60 = 376.8 \text{ cm}\).
- Now, calculate the distance each hand travels in 30 minutes:
- The second hand completes one full rotation every minute, so in 30 minutes, it completes 30 rotations. The total distance \(D_s\) traveled is: \(D_s = 30 \times 471 = 14130 \text{ cm}\).
- The minute hand completes one full rotation every 60 minutes. In 30 minutes, it completes half a rotation. The total distance \(D_m\) traveled is: \(D_m = 0.5 \times 376.8 = 188.4 \text{ cm}\).
- Calculate the additional distance the second hand travels over the minute hand:
- The extra distance \(x\) is given by: \(x = D_s - D_m = 14130 - 188.4 = 13941.6 \text{ cm}\).
- Convert this value to meters by dividing by 100: \(x = \frac{13941.6}{100} = 139.416 \text{ meters}\).
Thus, the value of \(x\) in meters is nearly 139.4 meters.
Approach Solution -2
The distance traveled by the tip of the minute hand in one revolution is:
\( x_{\text{min}} = \pi \times r_{\text{min}} = \pi \times \frac{60}{100} \, \text{m}. \)
\( x_{\text{min}} = 3.14 \times 0.6 = 1.884 \, \text{m}. \)
The distance traveled by the tip of the second hand in 30 minutes is:
\( x_{\text{second}} = 30 \times 2\pi \times r_{\text{second}} \)
\( x_{\text{second}} = 30 \times 2 \times 3.14 \times \frac{75}{100} \, \text{m}. \)
\( x_{\text{second}} = 30 \times 4.71 = 141.3 \, \text{m}. \)
The difference in distance is:
\( x = x_{\text{second}} - x_{\text{min}} = 141.3 - 1.884 \, \text{m}. \)
\( x = 139.4 \, \text{m}. \)
Final Answer: 139.4 m.
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