Tim and rick both can run at speed vr and walk at speed vw, with vr> vw. they set off together on a journey of distance d. rick walks half of the distance and runs the other half. tim walks half of the time and runs the other half.

part a
how long does it take rick to cover the distance d?
express the time taken by rick in terms of vr, vw, and d.

part b
find rick's average speed for covering the distance d.
express rick's average speed in terms of vr and vw.

part c
how long does it take tim to cover the distance?
express the time taken by tim in terms of vr, vw, and d.

Answers:

Part A: T = [ D/2 ] [ 1 / Vr + 1 / Vw) ]

Part B: Average speed = 2 × Vr × Vw / [ Vr + Vw ]

Parte C: T = 2 D / [Vr + Vw]

1) Variables:

D = total distance

Dr = run distance

Dw = walk distance

Vr = run speed

Vw = walk speed

T = total time

Tr = run time

Tw = walk time

2) Formulas:

i) velocity = distance / time

ii) distance = velocity × time

iii) time = distance / velocity

iv) Average speed = total distance / total time

3) General table:

person run walk

RicK D/2 D/2

Tim T /2 T /2

4) Part A. How long does it take Rick to cover the distance D?

i) T = Tr + Tw

ii) Tr = Dr / Vr

iii) Dr = D / 2

⇒ Tr = [D/2] / Vr = D / [ 2Vr]

iv) Tw = [D/2] / Vw = D / [2Vw]

v) T = Tr + Tw = [D / 2] × [ 1 / Vr + 1 / Vw ]

T = [ D/2 ] [ 1 / Vr + 1 / Vw) ] ← answer

5) Part B. Find Rick's average speed for covering the distance D.

Average speed = D / T = D × 1 / T = D × [2 / D] × [Vr×Vw] / [Vr + Vw]

Average speed = 2 × Vr × Vw / [ Vr + Vw ]

5) Part C How long does it take Tim to cover the distance?

i) Tr = Dr / Vr = T / 2 ⇒ Dr = Vr × T /2

ii) Tw = Dw / Vw = T / 2 ⇒ Dw = Vw × T /2

iii) ⇒ D = Dr + Dw = Vr × T / 2 + Vw × T / 2

⇒ D = [Vr + Vw] × T / 2

iv) ⇒ T = 2 D / [Vr + Vw] ← answer.

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