Worked connection for MMP-SUFFICIENT-ANSWER

1. Make the receiving question specific. A supplied account has two nonnegative populations with laws x’=-x and y’=-2y, where time uses the unit in which these rates are stated. The initial total is 1, but its split is unknown. Is the total remaining at time 3 below 0.1? This case starts with those laws; establishing a physical or other subject law is a different contribution.

2. Recover the omitted contribution. With z=x+y, MMP.9 gives z’=-x-2y. The same z=1 can have derivative -1 or -2, so the current total alone does not determine its rate of change. This result identifies the missing distinction and supplies alternatives for a bounding calculation.

3. Derive an answer shared by those alternatives. Put x(0)=a and y(0)=1-a, with 0≤a≤1. The equations give z(t)=a*exp(-t)+(1-a)*exp(-2t). MATH.20 uses this convex combination to derive exp(-2t)≤z(t)≤exp(-t) for t≥0. At time 3, exp(-3)<0.05<0.1. The upper bound settles the question for every initial split. No estimate of a is needed for this answer.

4. Return the consequence to the work. C.29 asks what the populations, rates, total and threshold represent and which assumptions permit the interpretation. Under the supplied account, the condition is satisfied at time 3. Using that consequence in a real decision also uses the subject premises that made these equations applicable.

5. Reopen what changes the answer. Suppose the next question is whether z(1)<0.2. The bounds straddle 0.2. The same formula reduces the decision to a<(0.2-exp(-2))/(exp(-1)-exp(-2)), approximately 0.278. An available bound on a may settle this. Obtaining more about a is useful when it can change the decision enough to justify its cost; C.11.DUA helps make that choice. Changed evolution laws instead return to step 2, since the old enclosure may fail.

A different population or physical decay process can use this example when it supplies the stated laws and interpretation. The reusable connection is broader: reduction exposes a lost distinction, a mathematical comparison bounds its effects, and the receiving question decides whether those effects need further work.