A person climbs from a Paris metro station to the street level by walking up a stalled escalator in 96.5 s. It takes 60.0 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?
To solve this problem, we need to figure out the relative speed between the person and the escalator.
Let's assume the distance from the metro station to the street level is "d."
When the escalator is not moving (stalled), the person's speed is equal to the speed of walking up the escalator, which we'll call "v."
When the escalator is moving, the person's actual speed is the difference between their walking speed and the escalator's speed, which we'll call "v'."
Now, let's set up the equations:
When the escalator is not moving:
Distance = speed x time
d = v x 96.5 seconds
When the escalator is moving:
Distance = speed x time
d = v' x t seconds
We are given that it takes 60 seconds to ride the same distance on the escalator when it's operating normally, so we can also write:
d = (v + v') x 60 seconds
Now, we can solve these equations to find the value of "t."
First, let's express v' in terms of v:
v' = v - (v/96.5) x 60
v' = v - 60v/96.5
v' = v - 0.619v
v' = 0.381v
Now, we equate the two expressions for "d":
v x 96.5 = (0.381v + v) x 60
96.5v = 0.381v x 60 + 60v x 60
96.5v = 0.381v x 60 + 3600v
96.5v - 3600v = 0.381v x 60
-3503.5v = 0.381v x 60
-3503.5 = 0.381 x 60
-3503.5/0.381 = x 60
-9178.477 = x 60
x = -9178.477/60
x ≈ -152.974
Since time cannot be negative, we ignore the negative sign. Therefore, it would take approximately 152.974 seconds (or rounded to 153 seconds) for the person to climb from the station to the street by walking up the moving escalator.
To find out how long it would take for the person to climb from the station to the street by walking up the moving escalator, we need to consider the difference in time it takes when the escalator is stalled versus when it is operating normally.
Let's assume that the distance from the station to the street is d.
When the escalator is stalled, it takes the person 96.5 seconds to climb up the distance d.
When the escalator is operating normally, it takes the person 60.0 seconds to ride up the distance d.
Since the person is walking up the moving escalator, their speed relative to the escalator is the difference between their speed when walking and the speed of the escalator.
Let's assume the speed of the escalator is v, and the person's walking speed is w.
When the person walks up the moving escalator, their speed relative to the ground is (w + v). So, to find the time it takes for the person to climb from the station to the street when walking up the moving escalator, we need to figure out the time it takes for the person to cover the distance d with a speed of (w + v).
To do this, we can set up the following equation:
d / (w + v) = 96.5
Solving this equation will give us the time it takes for the person to climb the distance d with a speed of (w + v) when the escalator is stalled.
Now, since the person takes 60.0 seconds to ride the distance d when standing on the operating escalator, we can set up another equation:
d / v = 60.0
Solving this equation will give us the time it takes for the person to cover the distance d with the speed of the escalator alone.
To find the total time it takes for the person to climb from the station to the street by walking up the moving escalator, we add the time calculated in the first equation (climbing with speed of w + v) to the second equation (climbing with speed of v alone):
d / (w + v) + d / v = total time
By substituting the known values for the time spent climbing when the escalator is stalled (96.5 seconds) and the time spent riding when the escalator is operating normally (60.0 seconds), we can solve for the total time.
This is just like a work problem. Think of the distance traveled in 1 second.
1/96.5 + 1/60 = 1/37
so, walking while riding takes only 37 seconds