Questions: A tank is being filled with a liquid. L(t), given below, is the amount of liquid in liters in the tank after t minutes. L(t)=1.25 t+61 Complete the following statements. Let L^(-1) be the inverse function of L Take x to be an output of the function L. That is, x=L(t) and t=L^(-1)(x). (a) Which statement best describes L^(-1)(x) ? The ratio of the amount of liquid (in liters) to the number of minutes, x. The reciprocal of the amount of liquid (in liters) after x minutes. The amount of time (in minutes) it takes to have x liters of liquid. The amount of liquid (in liters) after x minutes. (b) L^(-1)(x)=0 (c) L^(-1)(115)=

A tank is being filled with a liquid. L(t), given below, is the amount of liquid in liters in the tank after t minutes. L(t)=1.25 t+61 Complete the following statements. Let L^(-1) be the inverse function of L Take x to be an output of the function L. That is, x=L(t) and t=L^(-1)(x). (a) Which statement best describes L^(-1)(x) ? The ratio of the amount of liquid (in liters) to the number of minutes, x. The reciprocal of the amount of liquid (in liters) after x minutes. The amount of time (in minutes) it takes to have x liters of liquid. The amount of liquid (in liters) after x minutes. (b) L^(-1)(x)=0 (c) L^(-1)(115)=
Transcript text: A tank is being filled with a liquid. $L(t)$, given below, is the amount of liquid in liters in the tank after $t$ minutes. \[ L(t)=1.25 t+61 \] Complete the following statements. Let $L^{-1}$ be the inverse function of $L$ Take $x$ to be an output of the function $L$. That is, $x=L(t)$ and $t=L^{-1}(x)$. (a) Which statement best describes $L^{-1}(x)$ ? The ratio of the amount of liquid (in liters) to the number of minutes, $x$. The reciprocal of the amount of liquid (in liters) after $x$ minutes. The amount of time (in minutes) it takes to have $x$ liters of liquid. The amount of liquid (in liters) after $x$ minutes. (b) $L^{-1}(x)=0$ (c) $L^{-1}(115)=\square$
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Solution

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Solution Steps

Step 1: Determine the inverse function \( L^{-1}(x) \)

The function \( L(t) = 1.25t + 61 \) describes the amount of liquid in the tank after \( t \) minutes. To find the inverse function \( L^{-1}(x) \), we solve for \( t \) in terms of \( x \): \[ x = 1.25t + 61 \] Subtract 61 from both sides: \[ x - 61 = 1.25t \] Divide both sides by 1.25: \[ t = \frac{x - 61}{1.25} \] Thus, the inverse function is: \[ L^{-1}(x) = \frac{x - 61}{1.25} \]

Step 2: Answer part (a)

The inverse function \( L^{-1}(x) \) represents the amount of time (in minutes) it takes to have \( x \) liters of liquid in the tank. Therefore, the correct statement is:

  • The amount of time (in minutes) it takes to have \( x \) liters of liquid.
Step 3: Answer part (b)

We are asked to find \( L^{-1}(x) = 0 \). Using the inverse function: \[ L^{-1}(x) = \frac{x - 61}{1.25} = 0 \] Solve for \( x \): \[ \frac{x - 61}{1.25} = 0 \implies x - 61 = 0 \implies x = 61 \] Thus, \( L^{-1}(x) = 0 \) when \( x = 61 \).

Step 4: Answer part (c)

We are asked to find \( L^{-1}(115) \). Using the inverse function: \[ L^{-1}(115) = \frac{115 - 61}{1.25} = \frac{54}{1.25} = 43.2 \] Thus, \( L^{-1}(115) = 43.2 \).

Final Answer

(a) The amount of time (in minutes) it takes to have \( x \) liters of liquid.
(b) \( \boxed{x = 61} \)
(c) \( \boxed{L^{-1}(115) = 43.2} \)

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