Question 1209198
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Answers: 
Length = <font color=red>120 meters</font>
Speed = <font color=red>86.4 kilometers per hour</font> (equivalent to <font color=red>24 meters per second</font>)



Explanation:


L = length of the train


When the tip of the train engine is just starting to cross the bridge, the back-most part of the caboose is L meters away from the start of the bridge.
The caboose must travel this distance L, plus the length of the bridge, so that the entire train is off the bridge.
It might be useful to make a number line drawing.


The caboose travels L+1440 meters in 65 seconds.
distance = rate*time
rate = distance/time
rate = (L+1440 meters)/(65 seconds)
rate = (L+1440)/(65) meters per second


Using similar logic, we find that the rate can also be written as (L+1680)/75 when considering the tunnel scenario.


Those two rates are assumed to be equal since we assume the train doesn't speed up nor slow down when on the bridge vs tunnel.


bridge speed = tunnel speed
(L+1440)/65 = (L+1680)/75
75(L+1440) = 65(L+1680)
75L+108000 = 65L+109200
75L-65L = 109200-108000
10L = 1200
L = 1200/10
<font color=red>L = 120</font>


<font color=red>The train is 120 meters long</font>.


Then we use that length to find the train's speed
bridge speed = (L+1440)/65
bridge speed = (120+1440)/65
bridge speed = <font color=red>24 meters per second</font>
or,
tunnel speed = (L+1680)/75
tunnel speed = (120+1680)/75
tunnel speed = <font color=red>24 meters per second</font>
We get the same speed either way. 
You only need to show scratch work on one set rather than both. 
However, showing that both lead to the same speed helps confirm we got the correct length.


To convert from "meters per second" to "kilometers per hour", you could have this scratch work.
{{{
(matrix(1,2,24,"meters")/matrix(1,2,1,"second"))
(matrix(1,2,3600,"seconds")/matrix(1,2,1,"hour"))
(matrix(1,2,1,"kilometer")/matrix(1,2,1000,"meters"))
= 24*(3600/1000) =  matrix(1,4,86.4,"kilometers","per","hour")
}}}
Note that 1 hour = 60*60 = 3600 seconds.
We see that 24 meters per second = <font color=red>86.4 kilometers per hour</font>.
Each multiplier fraction is set up so that the "meters" and "seconds" units cancel.
The result 86.4 is exact and hasn't been rounded.
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