Question

napoleon's route through the alps involved travelling through the Grand st Bernard Pass, a climb of 6km and a descent of 4km. his army travelled twice as fast downhill as it did uphill and the whole journey took 8 hours. how long did it take him to reach the top?

Answer 1

It takes twice the time to travel a given distance while going uphill

compared to traveling downhill.

Response:

• The time it takes to reach the top is 6 hours

Which method can be used to calculate the duration of each part of the journey?

Given parameters are;

The distance to the top = 6 km

The length of the descent = 4 km

Duration of the whole journey, t = 8 hours

Speed the army travelled downhill = 2 × The speed while traveling uphill

Required:

How long it takes Napoleon to reach the top.

Solution:

Let v represent the speed while traveling uphill, we have;

The time it takes to travel uphill, t₁ = $\dfrac{6 \, km}{v}$

The time it takes to travel downhill, t₂ = $\mathbf{\dfrac{4 \, km}{2 \cdot v}}$

Duration of the journey, t = 8 = The total time = t₁ + t₂

Which gives;

$8 = \mathbf{\dfrac{6}{v} + \dfrac{4}{2 \cdot v}}$

$\dfrac{2 \times 6 + 4}{2 \cdot v} = 8$

8 × 2·v = 12 + 4

16·v = 16

$v = \dfrac{16}{16} = 1$

v = 1 km/hr

The time it takes to travel uphill, t₁ = $\dfrac{6 \, km}{1 \, km/hr}$ = 6 hours

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