A quick way of mixing drinks is to use a small amount of hot water to dissolve the powder. Measure how much powder you need and then slowly add the minimum amount of the hottest water you can get to dissolve it all. Then add cold water to fill it up. Check it before it finishes and stop adding water when the flavor is right. If you want to add ice, stop when it's a little stronger than you want because the ice will melt and dilute the drink.
Long, scientific explaination:
Sugar is usually the primary ingredient in these drinks (lemonade, iced tea) so lets look at how it interacts with water. We know that boiling water will dissolve 2.7 times more sugar than ice cold water, but in practice it seems to be more effective than 3 times as fast. That's because we are looking only at the thermodynamics (the end state). We know that all the sugar in the mix will dissolve completely so we don't need to worry about the thermodynamics, but we want to know is the kinetics behind it (the rate that it gets to the end state). We don't care about the final capacity, but how fast it can dissolve.
In terms of kinetics, we know that the free energy (delta G) increases in a non-linear rate with hot vs cold water. Practically what this means is the energy in the fast moving water molecules are breaking apart the sugar molecules so you don't have to through stirring. For example, 25 grams of sugar added to a liter of water will dissolve in 30 minutes in ice water, 15 minutes at room temperature, and 4 minutes in hot tap water. If you take near-boiling water from a water heater, it's over 10 times quicker to dissolve than cold water. It's even faster if you stir it as you add the water.
Usually about 2 cups of hot water would be enough for a 5 gallon cooler. If you use boiling water, then the final product will be 2 degrees warmer after adding cold water. A few ice cubes can cancel this out. It really does make mixing drinks more consistent and quicker.