Answer: The dependent variable is number of heat related illnesses
================================================
Step-by-step explanation:
Think of two dominoes made to stand on their ends, and they are close enough for one to knock over the other. The first domino will push the second one over, but not the other way around.
In that analogy, the first domino is the independent variable. It is free to do as it chooses. It can stand or fall over. The second domino must fall over if the first one dictates it; hence it corresponds to the dependent variable. It "depends" on the outcome of what happens with the first domino.
So to go back to the original problem, as temperature goes up, the number of heat illnesses goes up tells us that temperature is independent while the illness count is dependent on the temperature. Or you can think of it like this: if you increase the heat related illness count somehow, would that raise the environmental temperature? The answer is no. So that thought experiement shows that temperature can't be dependent.
Your book may have x be the temperature and y the number of illnesses. That way you can form an equation to connect the two variables.