A tank is being filled with a liquid. [tex]L(t)[/tex], given below, is the amount of liquid in liters in the tank after [tex]t[/tex] minutes.
[tex]\[L(t) = 1.25t + 61\][/tex]

Complete the following statements. Let [tex]L^{-1}[/tex] be the inverse function of [tex]L[/tex]. Take [tex]x[/tex] to be an output of the function [tex]L[/tex]. That is, [tex]x = L(t)[/tex] and [tex]t = L^{-1}(x)[/tex].

(a) Which statement best describes [tex]L^{-1}(x)[/tex]?
- The ratio of the amount of liquid (in liters) to the number of minutes, [tex]x[/tex].
- The reciprocal of the amount of liquid (in liters) after [tex]x[/tex] minutes.
- The amount of time (in minutes) it takes to have [tex]x[/tex] liters of liquid.
- The amount of liquid (in liters) after [tex]x[/tex] minutes.

(b) [tex]L^{-1}(x) = \square[/tex]

(c) [tex]L^{-1}(115) = \square[/tex]



Answer :

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.