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Math · Advanced Math
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Consider the following classic problem. A substance is 99% water. Some water evaporates, leaving a substance that is 98% water. How much of the water has evaporated? a. Getting the initial answer. Solve the given problem b. The classic status of the problem. Based on your answer to part a, indicate why you think this is called a classic problem. c. A numerical approach. In answering part a you may have used algebra, setting up unknowns and solving equations. Answer part a again, this time using no algebra, but only concrete, numerical reasoning. (For example, if the original substance had 1 unit of solid stuff and 99 units water, the evaporated substance still has 1 unit of solid stuff, so d. A diagrammatic approach. Solve the problem yet again, this time using a diagram as your representing the water and the solid in the original substance. A different rectangle can represent the evaporated substance e. Generalizing the problem. Solve the problem again, but this time replace the numerical values 99% and 98% in the statement of the problem with general parameters. (This is a first step to generalizing the problem.) f. A functions approach. The algebraic solution from part e is fully general, yet it is not fully revealing about why evaporating half the water has lowered the proportion of water only about 1%. In a sense, this solution is too general to focus on the essentials of this problem. Solve the problem again, but this time keep the specific numerical value 1% of the drop in water content part of the solution, and express the proportion of water evaporated as a function of the original proportion of water in the substance. g. Another functions approach. The approach outlined in the discussion of part f is not the only functions approach possible. You can also express the proportion of water evaporated as a function of the absolute amount of water in the substance. Try this. h. Summarize what you have learned from working on this problem analysis.

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