To solve these questions, let's first consider what we know about the function [tex]\( L(t) = 2.5t + 11.5 \)[/tex].
### (a) Describing [tex]\( L^{-1}(x) \)[/tex]
Given a function [tex]\( L(t) \)[/tex], its inverse function [tex]\( L^{-1}(x) \)[/tex] essentially reverses the input-output relationship. This means that if [tex]\( L(t) \)[/tex] gives us the amount of liquid in liters after [tex]\( t \)[/tex] minutes, [tex]\( L^{-1}(x) \)[/tex] will tell us the number of minutes it took to reach [tex]\( x \)[/tex] liters of liquid.
Thus, the correct statement that best describes [tex]\( L^{-1}(x) \)[/tex] is:
- The amount of time (in minutes) it takes to have [tex]\( x \)[/tex] liters of liquid.
### (b) Finding [tex]\( L^{-1}(x) \)[/tex]
To find the inverse function [tex]\( L^{-1}(x) \)[/tex]:
1. Start with the equation of [tex]\( L(t) \)[/tex]:
[tex]\[
x = 2.5t + 11.5
\][/tex]
2. Solve for [tex]\( t \)[/tex]:
[tex]\[
x - 11.5 = 2.5t
\][/tex]
[tex]\[
t = \frac{x - 11.5}{2.5}
\][/tex]
Hence, the inverse function is:
[tex]\[
L^{-1}(x) = \frac{x - 11.5}{2.5}
\][/tex]
### (c) Evaluating [tex]\( L^{-1}(90) \)[/tex]
Now we need to find the value of the inverse function when [tex]\( x = 90 \)[/tex]:
[tex]\[
L^{-1}(90) = \frac{90 - 11.5}{2.5}
\][/tex]
By performing the calculations, we get:
[tex]\[
L^{-1}(90) = \frac{78.5}{2.5} = 31.4
\][/tex]
### Summary of Results
(a) The best statement describing [tex]\( L^{-1}(x) \)[/tex] is:
- The amount of time (in minutes) it takes to have [tex]\( x \)[/tex] liters of liquid.
(b) The inverse function [tex]\( L^{-1}(x) \)[/tex] is:
[tex]\[
L^{-1}(x) = \frac{x - 11.5}{2.5}
\][/tex]
(c) Evaluating the inverse function at [tex]\( x = 90 \)[/tex]:
[tex]\[
L^{-1}(90) = 31.4
\][/tex]
Thus, the fully completed answer looks like this:
(a) The amount of time (in minutes) it takes to have [tex]\( x \)[/tex] liters of liquid.
(b) [tex]\( L^{-1}(x) = \frac{x - 11.5}{2.5} \)[/tex]
(c) [tex]\( L^{-1}(90) = 31.4 \)[/tex]
