`1 cm^(3)` of gold is drawn into a wire 0.1 mm in diameter. Find the length of the wire.

Explanation

To find the length of the wire made from 1 cm³ of gold with a diameter of 0.1 mm, we can follow these steps:

### Step-by-Step Solution:

1. **Convert the Volume of Gold to Cubic Meters:**
The volume of gold is given as 1 cm³. We need to convert this to cubic meters.
\[ 1 \text{ cm}^3 = 1 \times 10^{-6} \text{ m}^3 \]

**Hint:** Remember that \(1 \text{ cm} = 0.01 \text{ m}\), so \(1 \text{ cm}^3 = (0.01)^3 \text{ m}^3\).

2. **Determine the Radius of the Wire:**
The diameter of the wire is given as 0.1 mm. First, convert this to meters:
\[ 0.1 \text{ mm} = 0.1 \times 10^{-3} \text{ m} = 0.0001 \text{ m} \]
The radius \(r\) is half of the diameter:
\[ r = \frac{0.0001}{2} = 0.00005 \text{ m} \]

**Hint:** The radius is always half of the diameter.

3. **Calculate the Cross-Sectional Area of the Wire:**
The cross-sectional area \(A\) of the wire (which is cylindrical) can be calculated using the formula:
\[ A = \pi r^2 \]
Substituting the value of \(r\):
\[ A = \pi (0.00005)^2 = \pi \times 0.0000000025 \text{ m}^2 \]
Using \(\pi \approx \frac{22}{7}\):
\[ A \approx \frac{22}{7} \times 0.0000000025 \approx 7.14 \times 10^{-9} \text{ m}^2 \]

**Hint:** The area of a circle is calculated using the formula \(A = \pi r^2\).

4. **Relate the Volume of Gold to the Volume of the Wire:**
The volume of the wire can also be expressed as:
\[ \text{Volume} = \text{Area} \times \text{Length} \quad \Rightarrow \quad V = A \times L \]
Setting the volume of the wire equal to the volume of gold:
\[ 1 \times 10^{-6} = A \times L \]

**Hint:** The volume of a cylinder is given by the formula \(V = A \times L\).

5. **Solve for the Length \(L\):**
Rearranging the equation to find \(L\):
\[ L = \frac{1 \times 10^{-6}}{A} \]
Substituting the value of \(A\):
\[ L = \frac{1 \times 10^{-6}}{7.14 \times 10^{-9}} \approx 140.0 \text{ m} \]

**Hint:** When dividing, make sure to keep track of the powers of ten.

### Final Answer:
The length of the wire is approximately **140.0 meters**.