To solve this problem, we need to determine the radius of the wire that has been wound 30 times around a boiling tube. The total length of the windings is given as 9.3 mm.
### Step-by-Step Solution:
1. Convert the total length to centimeters:
Since the total length is given in millimeters and we often use centimeters for such measurements, we convert 9.3 mm to centimeters. We know that 1 cm = 10 mm. Therefore:
[tex]\[
\text{Total length in cm} = \frac{9.3 \text{ mm}}{10} = 0.93 \text{ cm}
\][/tex]
2. Identify that the total length of the wire is related to the circumference of each winding.
Each winding can be considered a circle with a certain radius, and the circumference of a circle is given by the formula:
[tex]\[
\text{Circumference} = 2 \pi \times \text{radius}
\][/tex]
3. Establish the relationship between the total length and the number of windings:
Given that the wire is wound 30 times, the total length of the wire is the sum of the circumferences of all windings. Thus:
[tex]\[
\text{Total length} = \text{Number of windings} \times \text{Circumference of one winding}
\][/tex]
Plugging in the values we have:
[tex]\[
0.93 \text{ cm} = 30 \times (2 \pi \times \text{radius})
\][/tex]
4. Solve for the radius:
To find the radius, we rearrange the formula to solve for the radius:
[tex]\[
\text{radius} = \frac{\text{Total length}}{30 \times 2 \pi}
\][/tex]
Substitute the known values:
[tex]\[
\text{radius} = \frac{0.93 \text{ cm}}{30 \times 2 \pi}
\][/tex]
We know that [tex]\(\pi \approx 3.14159\)[/tex].
5. Numerical solution:
Plugging in the value of [tex]\(\pi\)[/tex]:
[tex]\[
\text{radius} = \frac{0.93}{30 \times 2 \times 3.14159} \approx \frac{0.93}{188.4955}
\][/tex]
Performing the division gives:
[tex]\[
\text{radius} \approx 0.0049338 \text{ cm}
\][/tex]
### Summary:
- The total length of the wire in centimeters is [tex]\(0.93 \, \text{cm}\)[/tex].
- The radius of the wire is approximately [tex]\(0.0049338 \, \text{cm}\)[/tex] or [tex]\(4.9338 \, \text{mm}\)[/tex].
Therefore, we have:
[tex]\[
\boxed{0.93 \text{ cm}, \ 0.0049338 \text{ cm}}
\][/tex]
which implies that the radius of the wire is approximately [tex]\(0.49338 \, \text{mm}\)[/tex].
