In the sentence: "Cooling needs are expected to increase rapid with rising incomes and temperatures," identify the most appropriate replacement for the highlighted phrase to make the sentence grammatically correct and contextually meaningful.

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**Solution:**

1. The phrase "increase rapid" is grammatically incorrect because "rapid" is an adjective, but here an adverb is required to modify the verb "increase".

2. The correct adverb form of "rapid" is "rapidly".

3. Among the options, "to increase rapidly with rising" correctly uses the infinitive verb form "to increase" followed by the adverb "rapidly" and the phrase "with rising incomes and temperatures," making the sentence grammatically sound and contextually appropriate.

4. Other options either misuse verb forms or adverbs, or are grammatically incorrect.

**Answer:** to increase rapidly with rising

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