Final answer:
The constant of proportionality is a rate at which two quantities are related. In a cooking example with pancakes, it's the ratio of ingredients in the recipe. In a physics example with Hooke's Law, it's the spring's force constant that relates force to displacement.
Step-by-step explanation:
The constant of proportionality in Austen's cereal box storage context is a mathematical concept that relates one quantity to another at a constant rate. For example, if we're analyzing a recipe for pancakes where 2 cups (c) of pancake mix is equivalent to 1 egg, and also equivalent to 1/2 c of milk, the constant of proportionality would be the ratio that exists between each pair of ingredients for a single recipe. In this case, we could establish ratios such as 2 c of pancake mix to 1 egg or 4 c of pancake mix to 2 eggs, where the proportionality constant remains the same (2/1 for pancake mix to eggs).
Similarly, in a physics context, the constant of proportionality can refer to other relationships like Hooke's Law, which describes the force (F) needed to extend or compress a spring by a distance (x) proportional to that distance, represented by F = kx, where k is the spring's force constant. If a spring stretches 8.00 cm for a 10.0 kg load, we would solve for k by rearranging the equation to k = F/x. Assuming that gravity is 9.8 m/s2, the force due to the 10.0 kg load will be 10 kg * 9.8 m/s2 = 98 N. Converting 8.00 cm to meters gives 0.08 m, so k = 98 N / 0.08 m = 1225 N/m, which is the spring's force constant.