To solve this problem, we need to determine the percentage of river water that can be used for cooling if the temperature of the water returning to the river cannot increase by more than 10.0 K.

Given:

• Initial temperature of river water ($T_{\text{initial}}$) = 290 K
• Final temperature of returning water ($T_{\text{return}}$) = 320 K
• Maximum allowable temperature increase = 10.0 K
• Specific heat capacity of water ($c$) = 4.184 J/g·K

Approach:

1. Determine the temperature increase of the returned water: The returned water's temperature increases by: $\Delta T = T_{\text{return}} - T_{\text{initial}} = 320 \, \text{K} - 290 \, \text{K} = 30 \, \text{K}$

   So, the actual temperature increase is 30 K.

2. Required maximum temperature increase: By law, the temperature increase is allowed to be no more than 10.0 K.

3. Calculate the percentage of water that can be used: Let $x$ be the percentage of river water used for cooling. Since the actual temperature increase is 30 K but the allowable increase is only 10 K, we can set up a proportion: $\frac{\Delta T_{\text{allowed}}}{\Delta T_{\text{actual}}} = \frac{x}{100}$ Substituting values: $\frac{10 \, \text{K}}{30 \, \text{K}} = \frac{x}{100}$ Solving for $x$: $x = \frac{10}{30} \times 100 = 33.33\%$

Conclusion:

Only 33.33% of the river water can be used for cooling to comply with the legal requirement that the temperature increase of the returned water does not exceed 10.0 K.

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Would you like more details or have any questions?

Here are some related questions you might be interested in:

1. How is specific heat capacity related to temperature change?
2. What is the effect of increasing the flow rate of cooling water?
3. How does industrial cooling affect aquatic ecosystems?
4. What are alternative cooling methods that reduce water temperature increases?
5. How do laws regulating water temperature affect industrial processes?

Tip: Specific heat capacity measures how much energy is required to change the temperature of a substance by a given amount.