Problem 82 - Algebra and Trigonometry

Question

Toricelli's Law A tank holds 100 gallons of water, which drains from a leak at the bottom, causing the tank to empty in 40 minutes. Toricelli's Law gives the volume of water remaining in the tank after \(t\) minutes as $$ V(t)=100\left(1-\frac{t}{40}\right)^{2} $$ (a) Find \(V^{-1} .\) What does \(V^{-1}\) represent? (b) Find \(V^{-1}(15) .\) What does your answer represent?

Step-by-Step Solution

Answer

(a) \( V^{-1}(V) = 40\left(1 - \sqrt{\frac{V}{100}}\right) \). (b) \( V^{-1}(15) \approx 24.52 \); it represents the time to reach 15 gallons.

Step 1: Understanding Toricelli's Law Function

Toricelli's Law gives us the function \( V(t) = 100\left(1-\frac{t}{40}\right)^2 \), which describes the volume of water remaining after \( t \) minutes. \( V(t) \) is the remaining volume at a given time.

Step 2: Understanding the Inverse Function \( V^{-1} \)

The inverse function \( V^{-1} \) will give us the time \( t \) needed for a certain volume of water \( V \) to be remaining in the tank. Thus, \( V^{-1}(V(t)) = t \). Our goal is to find \( V^{-1} \) such that we can derive \( t \) as a function of \( V \).

Step 3: Finding the Inverse Function \( V^{-1} \)

To find \( V^{-1} \), solve for \( t \) in terms of \( V \). Start from the equation \( V = 100\left(1-\frac{t}{40}\right)^2 \).\Divide both sides by 100: \( \frac{V}{100} = \left(1-\frac{t}{40}\right)^2 \).\Take the square root of both sides: \( \sqrt{\frac{V}{100}} = 1-\frac{t}{40} \).\Solve for \( t \): \( \frac{t}{40} = 1 - \sqrt{\frac{V}{100}} \).\Thus, \( t = 40\left(1 - \sqrt{\frac{V}{100}}\right) \). This is \( V^{-1}(V) \).

Step 4: Evaluate \( V^{-1}(15) \)

Now substitute \( V = 15 \) into the expression for \( V^{-1} \):\\( t = 40\left(1 - \sqrt{\frac{15}{100}}\right) \).\Calculate \( \sqrt{\frac{15}{100}} = \sqrt{0.15} \approx 0.387 \).\Thus, \( t \approx 40(1 - 0.387) \approx 40(0.613) \approx 24.52 \). So, \( V^{-1}(15) = 24.52 \) minutes.

Step 5: Interpreting \( V^{-1} \) and \( V^{-1}(15) \)

\( V^{-1} \) represents the time in minutes it takes for the water level to reach a specific volume \( V \). \( V^{-1}(15) \approx 24.52 \) means it takes approximately 24.52 minutes for the tank to be left with 15 gallons of water.

Key Concepts

Volume of a Tank

Understanding the volume of a tank is essential in applying Toricelli's Law. When we talk about the volume of water in the tank, we refer to the quantity of space that the water occupies. In this case, the function provided by Toricelli's Law gives us this volume over time as the water drains. The volume function, given by \[ V(t) = 100\left(1-\frac{t}{40}\right)^2 \]shows us that the volume decreases quadratically over time.

• Initially, when no time has passed (\(t = 0\)), the volume is 100 gallons.
• As time progresses to 40 minutes, the function indicates that the volume reduces to 0 gallons, as expected when the tank is fully drained.
  

This function helps us model and predict how the volume changes, which can be particularly helpful for planning and understanding draining processes.

Inverse Function

An inverse function reverses the roles of inputs and outputs from the original function. For the given volume function \(V(t) = 100\left(1-\frac{t}{40}\right)^2\), the inverse function \(V^{-1}(V)\) provides the time \(t\) required for the tank to reach a specified volume \(V\).To find the inverse function, we solve for \(t\) in terms of \(V\), which leads to: \[t = 40\left(1 - \sqrt{\frac{V}{100}}\right)\]

• The process involves isolating \(t\) by mathematical operations including dividing, taking square roots, and solving equations linearly (all further explained in this text).
  

This inverse function is valuable as it allows for determining the time needed for any specified remaining volume, expanding the usefulness of the original volume function.

Solving Equations

Solving equations is a fundamental aspect of mathematics that allows for finding unknown values based on given relationships. When solving for the inverse function, we start with:\[ V = 100\left(1-\frac{t}{40}\right)^2 \]

• First, divide both sides by 100 to simplify the equation.
  
• Next, take the square root of both sides to eliminate the square.
• Finally, solve linearly for \(t\) by involving simple arithmetic operations.
  

By following these steps, we isolate \(t\), resulting in the inverse function. This process exemplifies using algebraic manipulation techniques to derive useful mathematical expressions, crucial in real-world problem-solving scenarios.

Mathematical Modeling

Mathematical modeling involves creating equations to represent real-world systems. In the context of Toricelli's Law, the volume function models the process of a tank draining over time.

• This particular model, given by \( V(t) = 100\left(1-\frac{t}{40}\right)^2 \), effectively captures the non-linear nature of the draining process.
  
• The quadratic form indicates how the rate of volume decrease changes over time, a critical realization of Toricelli's Law.
• By finding an inverse, we further enhance this model's capability, making predictions based on desired outcomes, such as remaining water volume.
  

Utilizing such models allows scientists and engineers to predict behaviors of systems and make informed decisions, demonstrating the power and utility of mathematical modeling across disciplines.