Question

Suppose a railroad rail is 4 kilometers and it expands on a hot day by 16 centimeters in length. Approximately how many meters would the center of the rail rise above the​ ground?

Answer 1

On a hot day, the center of the railroad rail would rise approximately 8 millimeters above the ground.

Step-by-step explanation:

The expansion of the railroad rail can be calculated using the formula:

`\[ \text{Expansion} = \text{Coefficient of Expansion} * \text{Original Length} * \text{Change in Temperature} \]`

In this case, the coefficient of linear expansion for steel (commonly used for railroad rails) is approximately
`\(0.000012/\degree C\)`, the original length of the rail is 4 kilometers (or 4000 meters), and the change in temperature is the equivalent of 16 centimeters (or 0.16 meters). Plugging these values into the formula:

`\[ \text{Expansion} = 0.000012 * 4000 * 0.16 \]`

`\[ \text{Expansion} = 0.768 \, meters \]`

This is the total expansion of the rail. However, we are interested in the rise of the center, which is half of the total expansion. Therefore, the rise of the center is:

`\[ \text{Rise of Center} = 0.5 * 0.768 \]`

`\[ \text{Rise of Center} = 0.384 \, meters \]`

To convert this into millimeters, we multiply by 1000:

`\[ \text{Rise of Center} = 384 \, millimeters \]`

So, on a hot day, the center of the railroad rail would rise approximately 8 millimeters above the ground. This expansion due to temperature changes is crucial to consider in engineering and construction to prevent issues such as buckling or warping of materials.

Answer 2

17.8 meters

Step-by-step explanation:

Given:

before expansion railroad rail= 4 km

expansion= 16 cm

railroad makes an isosceles triangle at the center of the rail when it expands

The base of the triangle=4 km

As 1km=100,000 cm

thus 16 cm= 16/100,000 km

each side of this triangle= (4 + 16/100,000) / 2 = 4.0001 / 2 = 2.00008 km

One half of the isosceles triangle makes a right triangle with one of the side =4 / 2 = 2 km

and the hypotenuse = 2.00008 km

Using Pythagoras theorem to find the third side of this right angles triangle:

c^2=a^2+b^2

2.00008^2= 2^2 + b^2

2.00008^2 - 2^2=b^2

b=0.0178 km

b= 17.8 m

hence, approximately 17.8 meters would the center of the rail rise above the​ ground!