Twelve-Week Running Plan: Arithmetic vs Rounded Geometric Growth
A runner begins a 12-week programme. In week 1 she runs 18 km. Plan A increases weekly distance by 2.5 km each week. Plan B increases by 8% each week, rounded to the nearest 0.1 km at the start of each week before running. She will follow one plan for 12 weeks. Her coach says a plan is acceptable only if total distance is at least 300 km and no single week exceeds 45 km. Compare total distances, identify first week each plan exceeds 45 km, and decide which plan meets both conditions. Use exact arithmetic where possible and rounded weekly values only when the plan requires rounding.
1. Compute the 12-week total and the maximum weekly distance for each plan. State the first week, if any, when the weekly distance exceeds 45 km.
Read the worked answer
Plan A is arithmetic with a1=18 and d=2.5. Its 12-week total is (12/2)(2·18+11·2.5)=6(36+27.5)=6·63.5=381 km. Week 12 is 18+11·2.5=45.5 km, so Plan A first exceeds 45 km in week 12.
Plan B rounded weekly distances are: 18.0, 19.4, 21.0, 22.7, 24.5, 26.5, 28.6, 30.9, 33.4, 36.1, 39.0, 42.1 km. Their sum is 342.2 km. The largest is 42.1 km, so Plan B never exceeds 45 km.
2. Which plan meets the coach's two conditions as written? Justify using both the total and the weekly cap.
Read the worked answer
Plan A has total 381 km, which is at least 300 km, but week 12 is 45.5 km, which violates the rule that no single week exceeds 45 km. Therefore Plan A is not acceptable as written.
Plan B has total 342.2 km, which is at least 300 km, and its maximum week is 42.1 km, which is at most 45 km. Therefore Plan B is the only plan that meets both conditions.
3. If the coach allows week 12 of Plan A to be capped at 45 km, how do the totals compare, and does the choice change?
Read the worked answer
Capping week 12 of Plan A reduces its total by 45.5−45.0=0.5 km, so Plan A becomes 381−0.5=380.5 km. Plan B remains 342.2 km because no cap is needed.
With the cap, Plan A now satisfies the weekly limit and still has the larger total. The choice changes from Plan B to Plan A if the cap is allowed. If the 45 km limit is a strict rule that cannot be modified, Plan B remains the only acceptable original plan.