The depth of a pond is 180 cm and is being reduced by 1 cm per week. The depth of a second pond is 160 cm and is being reduced by `1/2` cm per week. If the depths of both ponds continue to be reduced at these constant rates, in about how many weeks will the ponds have the same depth?

Options

A. 10
B. 20
C. 40
D. 80

Answer

To solve the problem of when the two ponds will have the same depth, we can follow these steps:

1. **Define the Variables**:
- Let \( x \) be the number of weeks after which both ponds will have the same depth.

2. **Depth of the First Pond**:
- The initial depth of the first pond is 180 cm.
- The depth decreases by 1 cm per week.
- Therefore, after \( x \) weeks, the depth of the first pond will be:
\[ \text{Depth of First Pond} = 180 - x \]

3. **Depth of the Second Pond**:
- The initial depth of the second pond is 160 cm.
- The depth decreases by \( \frac{1}{2} \) cm (or 0.5 cm) per week.
- Therefore, after \( x \) weeks, the depth of the second pond will be:
\[ \text{Depth of Second Pond} = 160 - 0.5x \]

4. **Set the Depths Equal**:
- To find when both ponds have the same depth, we set the two equations equal to each other:
\[ 180 - x = 160 - 0.5x \]

5. **Solve for \( x \)**:
- Rearranging the equation gives:
\[ 180 - 160 = -0.5x + x \]
- Simplifying further:
\[ 20 = 0.5x \]
- To isolate \( x \), multiply both sides by 2:
\[ 40 = x \]

6. **Conclusion**:
- Therefore, it will take approximately 40 weeks for the depths of both ponds to be the same.