Out-and-back circuits and changing growth rates

Fig. 1: supplied circuit points O, P₁, P₂, P₃, P₄ and distances 50 m, 100 m, 300 m, 900 m.OP₁P₂P₃P₄50 m100 m300 m900 mFig. 1
Question diagram 1
Fig. 1: supplied circuit points O, P₁, P₂, P₃, P₄ and distances 50 m, 100 m, 300 m, 900 m.OP₁P₂P₃P₄50 m100 m300 m900 mFig. 1

Fig. 1: supplied circuit points O, P₁, P₂, P₃, P₄ and distances 50 m, 100 m, 300 m, 900 m.

Besto’s coach designs a training programme starting with a 400m run on session 1. Each subsequent session covers 250m more than the previous.

(i) Find the minimum sessions for at least 20km in one session. marks

For another group in Besto, a stamina circuit starts at O with points P1,P2,P3,…P_1,P_2,P_3,\ldots increasingly far away on a straight line. Stage 1 goes O to P1P_1 and back, stage 2 O to P2P_2 and back, etc. The diagram (Fig. 1) gives OP1=50OP_1=50m, P1P2=100P_1P_2=100m, P2P3=300P_2P_3=300m and PnPn+1=3Pn−1PnP_nP_{n+1}=3P_{n-1}P_n.

(ii) Find distance after nn stages. marks

(iii) Hence find distance from O and travel direction after exactly 42km. marks

Choco’s coach designs another programme starting at 400m in session 1. Each subsequent distance increases by 10% of the previous session’s distance. From session 11 onward and for every subsequent session, distance increases by rr% of the previous distance.

(iv) If Choco runners cover at least 20km in session 70, find the range of rr. marks

[2022/SAJC/Prelim/I/9]

Total marks: 13

Solution

(i)

Session reaching 20 km

Find the first Besto session whose distance is at least 20 km.

Hint

  1. Use the AP term formula, not the AP sum.

Answer · (i)

Session 8080.

(ii)

Distance after n circuit stages

Find the distance run after completing nn circuit stages.

Hint

  1. First derive OPj from the gaps; then double it for the return journey.

Answer · (ii)

50(3n−1) m50(3^n-1)\,\mathrm m.

(iii)

Position and direction after 42 km

After exactly 42 km, find the runner’s distance from O and direction.

Explore this idea

Hint

  1. A stage total includes the return leg; check whether the remaining distance reaches its turning point.

Answer · (iii)

5600 m5600\,\mathrm m from O, travelling away.

Explore the idea

Find the last completed circuit stage

Each complete stage ends back at O. Compare the cumulative distance with 42000 m before considering the partial next stage.

Completed stagesn = 6
Try an example
completed distance (m) at n = 6
36400.0
Key idea

Six stages cover 36400 m. The remaining 5600 m lies on stage 7’s outward leg.

Ideas to try
Compare the values on either side of the boundary. What makes one count qualify?
(iv)

Required new percentage growth

Find the range of new percentage increases rr for Choco to reach at least 20 km in session 70.

Hint

  1. Count transitions between session indices, not the number of labelled sessions.

Answer · (iv)

r≥100[(50/1.19)1/60−1]≈5.22217%r\ge100[(50/1.1^9)^{1/60}-1]\approx5.22217\%.

The rounded 5.22% is approximate; retain the exact lower bound.

Takeaway

Count stages and growth transitions separately

Sessions 1–10 contain nine 10% increases. Session 11 receives the first new-rate increase, leaving 60 new multiplications through session 70.