Certainly! Let's break down the problem step-by-step to determine the radius of the wire, given that the thin wire was wound 30 times around a boiling tube, with a total length of 0.3 mm.
### Step-by-Step Solution:
1. Understand the Problem:
- You have a thin wire wound 30 times around a tube.
- The total length of the wire is 0.3 mm.
- You need to find the radius of the wire.
2. Establish Known Values:
- Total length of the wire (L) = 0.3 mm
- Number of windings (N) = 30
3. Determine the relationship between the total length and the circumference:
- When a wire is wound in a circular way, the total length of the wire is the sum of the circumference of all windings.
- The circumference of one winding is given by [tex]\( 2 \pi r \)[/tex], where [tex]\( r \)[/tex] is the radius of one winding.
4. Form the Equation:
- The total length of the wire is the product of the number of windings and the circumference of one winding:
[tex]\[
L = N \cdot (2 \pi r)
\][/tex]
Therefore:
[tex]\[
0.3 \, \text{mm} = 30 \cdot (2 \pi r)
\][/tex]
5. Solve for the Radius (r):
[tex]\[
0.3 = 30 \cdot 2 \pi r
\][/tex]
[tex]\[
0.3 = 60 \pi r
\][/tex]
[tex]\[
r = \frac{0.3}{60 \pi}
\][/tex]
6. Calculate the Value of the Radius:
Now, using the constant value of [tex]\( \pi \approx 3.14159 \)[/tex]:
[tex]\[
r = \frac{0.3}{60 \pi} \approx \frac{0.3}{188.495}
\][/tex]
[tex]\[
r \approx 0.00159154943 \, \text{mm}
\][/tex]
### Conclusion:
Thus, the radius of the wire is approximately 0.00159154943 mm.
This step-by-step process aligns with the methodology needed to solve the problem and finds the radius of the wire correctly.
