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The linear thermal expansion is the tendency to change in length of an object in response to a change in temperature.

The change in the length of the object is given as

\(\Delta l = \alpha {l_0}\Delta T\).

Here, \(\alpha \) is the coefficient of linear expansion.

The radius of the Earth is \(R = 6.38 \times {10^6}\;{\rm{m}}\).

The initial temperature of the steel band is \(T = 25^\circ {\rm{C}}\).

The temperature of the steel band after heating is \(T' = 55^\circ {\rm{C}}\).

The coefficient of linear expansion for steel is \(\alpha = 12 \times {10^{ - 6}}\;{\left( {{\rm{C^\circ }}} \right)^{ - 1}}\).

Since the steel band was initially fit snugly around the Earth鈥檚 equator, the length of the steel band would be equal to the circumference of the Earth鈥檚 equator.

The initial length of the steel band is

\({l_0} = 2\pi R\).

The change in the length of the steel band on heating is

\(\begin{aligned}\Delta l &= \alpha {l_0}\Delta T\\ &= 2\pi \alpha R\Delta T\end{aligned}\).

Substitute the values into the above expression.

\(\begin{aligned}\Delta l &= 2\pi \left( {12 \times {{10}^{ - 6}}{{\left( {{\rm{C}}^\circ } \right)}^{ - 1}}} \right)\left( {6.38 \times {{10}^6}\;{\rm{m}}} \right)\left( {55^\circ {\rm{C}} - 25^\circ {\rm{C}}} \right)\\ &= 4593.6\pi \;{\rm{m}}\end{aligned}\)

The change in length of the band results in the change in its radius. If\(\Delta R\)is the change in the radius, then change in length can also be written in the following manner:

\(\begin{aligned}\Delta l &= 2\pi \Delta R\\\Delta R &= \frac{{\Delta l}}{{2\pi }}\end{aligned}\)

Substitute the values into the above equation.

\(\begin{aligned}\Delta R &= \frac{{4593.6\pi \;{\rm{m}}}}{{2\pi }}\\ &= 2296.8\;{\rm{m}} \times \frac{{1\;{\rm{km}}}}{{{{10}^3}\;{\rm{m}}}}\\ &\approx 2.3\;{\rm{km}}\end{aligned}\)

Thus, the height above the Earth would be equal to the change in radius, and it is equal to \(2.3\;{\rm{km}}\).