Sure, let's work through the problem step-by-step.
Given the function [tex]\(L(t)\)[/tex], which describes the amount of liquid in liters in the tank after [tex]\(t\)[/tex] minutes:
[tex]\[ L(t) = 1.25t + 61 \][/tex]
We need to address three parts:
### Part (a)
We need to describe the inverse function [tex]\(L^{-1}(x)\)[/tex]. Given that [tex]\(L(t)\)[/tex] outputs the amount of liquid in liters after a certain time [tex]\(t\)[/tex], the inverse function [tex]\(L^{-1}(x)\)[/tex] will take the amount of liquid [tex]\(x\)[/tex] and give the time [tex]\(t\)[/tex].
Out of the provided options, the best description for the inverse function [tex]\(L^{-1}(x)\)[/tex] is:
"The amount of time (in minutes) it takes to have [tex]\(x\)[/tex] liters of liquid."
### Part (b)
We need to find the expression for [tex]\(L^{-1}(x)\)[/tex].
Starting from the original function:
[tex]\[ x = L(t) = 1.25t + 61 \][/tex]
To find the inverse, solve for [tex]\(t\)[/tex]:
[tex]\[ x - 61 = 1.25t \][/tex]
[tex]\[ t = \frac{x - 61}{1.25} \][/tex]
Thus, the inverse function is:
[tex]\[ L^{-1}(x) = \frac{x - 61}{1.25} \][/tex]
### Part (c)
Finally, we need to calculate [tex]\(L^{-1}(115)\)[/tex], which is the time it takes to have 115 liters of liquid in the tank.
Using the inverse function [tex]\(L^{-1}(x)\)[/tex]:
[tex]\[ L^{-1}(115) = \frac{115 - 61}{1.25} \][/tex]
Simplifying the expression:
[tex]\[ L^{-1}(115) = \frac{54}{1.25} \][/tex]
[tex]\[ L^{-1}(115) = 43.2 \][/tex]
So, the complete solutions are:
(a) The best statement to describe [tex]\(L^{-1}(x)\)[/tex] is: "The amount of time (in minutes) it takes to have [tex]\(x\)[/tex] liters of liquid."
(b) [tex]\(L^{-1}(x) = \frac{x - 61}{1.25}\)[/tex]
(c) [tex]\(L^{-1}(115) = 43.2\)[/tex]
These answers collectively provide the description and calculations pertaining to the given problem.
