Algebra-Oriented Solution

Problem for the homework 3, chapter 3.

You drive your car 5.0 km due East at 35km/hr, and suddenly realize that you forgot your wallet.  So, you return home, driving West at 40 km/hr, and upon arrival spend 10 minutes looking for it, and finally go back on the road for a total of 57.0 km due East.  If your average speed was 40 km/hr for the whole journey, what was your average speed on your last leg?

a)46 km/hr
b)49 km/hr
c)35 km/hr
d) 40 km/hr

2) Algebra-Oriented Solution

To East:
x₁=5km
V₁=35 km/h
t₁=x₁/V₁
Back to home, driving West:
V₂=40 km/h
t₂=x₁/V₂
Looking time t₃=10 minutes = 1/6 h
Distance of the last leg due East x₄ = 57 km
Distance Total (The total length of the total road pass) d = 2x₁+x₄
V_{average total}=V_A = 40km/h
V₄=?

Total time = (Distance Total)/ V_{average total} = d/V_A= (2x₁+x₄)/ V_A
Total time= t1+t2+t3+t4 = x₁/V₁ + x₁/V₂ + t₃ + x₄/V₄
(2x₁+x₄)/ V_A = x₁/V₁ + x₁/V₂ + t₃ + x₄/V₄
x₄/V₄  = (2x₁+x₄)/ V_A - x₁/V₁ - x₁/V₂ - t₃
V₄/ x₄  = 1/(  (2x₁+x₄)/ V_A - x₁/V₁ - x₁/V₂ - t₃   )
V₄  = x₄ /(  (2x₁+x₄)/ V_A - x₁/V₁ - x₁/V₂ - t₃   )

Mathematical Expression for Calculation:

x₁=5km, V₁=35 km/h, V₂=40 km/h, t₃=10 minutes, x₄ = 57 km, V_A = 40km/h

V₄  = x₄ /(  (2x₁+x₄)/ V_A - x₁/V₁ - x₁/V₂ - t₃   )

Expression to the Google calculator:
57km /(  (2* 5km+57km)/ 40 km/h – 5km/35km/h – 5km/40km/h - 10minutes   ) in km/h

Google result: 45.950096 km / hour

Answer: a) 46 km/h