Let's solve the problem step by step.
Given the function describing the relationship between the temperature [tex]\( T \)[/tex] (in degrees Celsius) and the height [tex]\( h \)[/tex] (in kilometers) above the surface:
[tex]\[ T(h) = 36.5 - 2.5h \][/tex]
(a) Which statement best describes [tex]\( T^{-1}(x) \)[/tex]?
To understand the inverse function [tex]\( T^{-1}(x) \)[/tex], recall that it reverses the roles of the input and output. This means that [tex]\( T^{-1}(x) \)[/tex] gives the height [tex]\( h \)[/tex] when the temperature is [tex]\( x \)[/tex] degrees Celsius.
Thus, the best description for [tex]\( T^{-1}(x) \)[/tex] is:
- The height above the surface (in kilometers) when the temperature is [tex]\( x \)[/tex] degrees Celsius.
(b) [tex]\( T^{-1}(x) = \)[/tex] [tex]$\square$[/tex]
To find the inverse function, we need to solve the equation [tex]\( T(h) = x \)[/tex] for [tex]\( h \)[/tex]:
[tex]\[ x = 36.5 - 2.5h \][/tex]
Rearrange this equation to solve for [tex]\( h \)[/tex]:
[tex]\[ 2.5h = 36.5 - x \][/tex]
[tex]\[ h = \frac{36.5 - x}{2.5} \][/tex]
So, the inverse function [tex]\( T^{-1}(x) \)[/tex] is:
[tex]\[ T^{-1}(x) = 14.6 - 0.4x \][/tex]
(c) [tex]\( T^{-1}(27) = \)[/tex] [tex]$\square$[/tex]
To find [tex]\( T^{-1}(27) \)[/tex], we substitute [tex]\( x = 27 \)[/tex] into the inverse function:
[tex]\[ T^{-1}(27) = 14.6 - 0.4 \times 27 \][/tex]
[tex]\[ T^{-1}(27) = 14.6 - 10.8 \][/tex]
[tex]\[ T^{-1}(27) = 3.8 \][/tex]
So, putting it all together:
(a) The best description of [tex]\( T^{-1}(x) \)[/tex] is:
- The height above the surface (in kilometers) when the temperature is [tex]\( x \)[/tex] degrees Celsius.
(b) [tex]\( T^{-1}(x) = 14.6 - 0.4x \)[/tex]
(c) [tex]\( T^{-1}(27) = 3.8 \)[/tex]
