Ten meter long steel railroad rails are laid end to end, with no space between, on a hot day when the temperature is 115°F. Six months later the temperature has dropped to -2°F.How much space now exists between each rail?

delta L/L = k delta T

delta L/L = 6.7*10^-6 inches/inch degree F (117 deg F)
L = 10 meters =394 inches
so
delta L = 394 * 6.7 *10^-6 * 117
= .309 inches = .784 cm = .0784 meters

To determine how much space exists between each rail after six months, we need to consider the change in temperature and the coefficient of thermal expansion of steel.

Here is how we can calculate it:

1. Determine the change in temperature:
The change in temperature is the difference between the initial and final temperatures.
ΔT = final temperature - initial temperature
ΔT = -2°F - 115°F
ΔT = -117°F

2. Find the coefficient of thermal expansion of steel:
The coefficient of thermal expansion (α) is a measure of how much a material expands or contracts with changes in temperature.
The average linear coefficient of thermal expansion for steel is approximately 11.7 x 10^-6 per °F.

3. Calculate the change in length for each rail:
The change in length (∆L) is given by the formula:
∆L = α * L * ΔT
Where:
- α is the coefficient of thermal expansion
- L is the initial length of the rail
- ΔT is the change in temperature

∆L = (11.7 x 10^-6 per °F) * (10 meters) * (-117°F)
∆L ≈ -0.013 meters

Since the rails were laid end to end with no space between initially, the negative change in length (∆L) means that they have contracted.

4. Calculate the space between each rail:
Since there were initially no spaces between the rails, the space between each rail after contraction is equal to the negative change in length (∆L).
Space between each rail = ∆L
Space between each rail ≈ -0.013 meters

Therefore, after six months, approximately -0.013 meters (or -13 millimeters) of space exists between each rail.