Let's solve the problem step-by-step.
1. Determine the total time: We need to convert the given time into seconds. The given time is 3 minutes and 45 seconds.
[tex]\[
\text{Total time in seconds} = 3 \times 60 + 45 = 180 + 45 = 225 \text{ seconds}
\][/tex]
2. Calculate the number of full circles: The second hand of the clock makes a full circle (360 degrees or [tex]\( 2\pi \)[/tex] radians) every 60 seconds. To find out how many full circles the second hand completes in the given time:
[tex]\[
\text{Number of full circles} = \frac{225 \text{ seconds}}{60 \text{ seconds/circle}} = 3.75 \text{ circles}
\][/tex]
3. Calculate the distance swept by the tip of the second hand: Each full circle corresponds to the circumference of the circle traced by the second hand. The circumference [tex]\( C \)[/tex] of a circle is given by:
[tex]\[
C = 2 \pi r
\][/tex]
where [tex]\( r \)[/tex] is the radius of the circle. In this case, the length of the second hand is the radius:
[tex]\[
r = 0.30 \text{ meters}
\][/tex]
4. Calculate the circumference:
[tex]\[
\text{Circumference} = 2 \times \pi \times 0.30 \text{ meters}
\][/tex]
5. Find the total distance swept: To find the total distance the tip of the second hand sweeps, we multiply the circumference by the number of full circles:
[tex]\[
\text{Distance swept} = 3.75 \times 2 \pi \times 0.30 \text{ meters}
\][/tex]
6. Evaluating the distance: Rounding to two decimal places, the distance swept by the tip of the second hand can be found as:
[tex]\[
\text{Distance swept} \approx 7.07 \text{ meters}
\][/tex]
Thus, the second hand of a clock with a length of 0.30 meters sweeps through a distance of approximately 7.1 meters in 3 minutes and 45 seconds.
Therefore, the correct answer is:
[tex]\[
\boxed{7.1 \text{ meters}}
\][/tex]
