Scientists have found a relationship between the temperature and the height above a distant planet's surface. $T(h)$, given below, is the temperature in Celsius at a height of $h$ kilometers above the planet's surface. The relationship is as follows.
\[
T(h)=48.5-2.5 h
\]
Complete the following statements.
Let $T^{-1}$ be the inverse function of $T$. Take $x$ to be an output of the function $T$.
That is, $x=T(h)$ and $h=T^{-1}(x)$.
(a) Which statement best describes $T^{-1}(x)$ ?
The ratio of the temperature (in degrees Celsius) to the number of kilometers, $x$.

The height above the surface (in kilometers) when the temperature is $x$ degrees Celsius.

The reciprocal of the temperature (in degrees Celsius) at a height of $x$ kilometers.
The temperature (in degrees Celsius) at a height of $x$ kilometers.
(b) $T^{-1}(x)=$
(c) $T^{-1}(33)=$

Step 1 ：The problem is asking for the inverse function of \(T(h)\), which is \(T^{-1}(x)\). The inverse function will give us the height above the surface when the temperature is \(x\) degrees Celsius.

Step 2 ：To find the inverse function, we need to switch the roles of \(h\) and \(x\) in the equation \(T(h) = 48.5 - 2.5h\), and then solve for \(h\).

Step 3 ：The inverse function of \(T(h)\) is \(T^{-1}(x) = 19.4 - 0.4x\). This function gives us the height above the surface when the temperature is \(x\) degrees Celsius.

Step 4 ：Now, we can use this function to find \(T^{-1}(33)\), which is the height above the surface when the temperature is 33 degrees Celsius.

Step 5 ：Final Answer: (a) The statement that best describes \(T^{-1}(x)\) is 'The height above the surface (in kilometers) when the temperature is \(x\) degrees Celsius.' (b) \(T^{-1}(x) = 19.4 - 0.4x\) (c) \(T^{-1}(33) = \boxed{6.2}\) kilometers.