Question

A person climbs from a Paris metro station to the street level by walking up a stalled escalator in 94 s. It takes \(66 \mathrm{s}\) to ride the same distance when standing on the escalator when it is operating normally. How long would it take for him to climb from the station to the street by walking up the moving escalator?

Short answer

Answer: It would take approximately 38.58 seconds for the person to climb from the station to the street by walking up the moving escalator.

Step-by-step solution

Assign variables

Let's assign variables for given data: - \(T_w\): the time it takes to walk up the stalled escalator, \(T_w=94\) s - \(T_r\): the time it takes to ride the same distance when the escalator is working, \(T_r=66\) s - \(T_\text{total}\): the time we need to find out for the person to climb from the station to the street by walking up the moving escalator

Calculate the distance

Since the distance covered in both cases is the same, we can denote this by the variable \(D\). To find this value, we can use the relationship: \(D = S_w \times T_w\) and \(D = S_e \times T_r\), where \(S_w\) is the person's walking speed and \(S_e\) is the escalator's speed. We can rewrite the equation as: \(S_w = \frac{D}{T_w}\) and \(S_e = \frac{D}{T_r}\)

Calculate the relative speed

When the person is walking up the moving escalator, their total speed would be the sum of their walking speed and the escalator's speed because they are moving in the same direction. We can write this as: \(S_\text{total} = S_w + S_e\) Substitute the values for speed from Step 2: \(S_\text{total} = \frac{D}{T_w} + \frac{D}{T_r}\)

Calculate the time it takes to climb by walking up the moving escalator

Now we can use the relationship distance = speed × time to find the total time taken by the person while walking up the moving escalator: \(T_\text{total} = \frac{D}{S_\text{total}}\) Substitute the expression for \(S_\text{total}\) from Step 3: \(T_\text{total} = \frac{D}{\frac{D}{T_w} + \frac{D}{T_r}}\) Note that we can cancel the \(D\) variable from the equation: \(T_\text{total} = \frac{1}{\frac{1}{T_w} + \frac{1}{T_r}}\) Plug in the values of \(T_w\) and \(T_r\): \(T_\text{total} = \frac{1}{\frac{1}{94\,\text{s}} + \frac{1}{66\,\text{s}}}\)

Calculate the final answer

We can now compute the value for \(T_\text{total}\): \(T_\text{total} = \frac{1}{\frac{94 + 66}{94 \times 66}} = \frac{94 \times 66}{94 + 66} = 38.579\) s Rounded to two decimal places, it would take approximately \(\boldsymbol{38.58}\) seconds for the person to climb from the station to the street by walking up the moving escalator.