Problem 62 - College Algebra with Modeling and Visualization

Question

A cylindrical tank contains 100 gallons of water. A plug is pulled from the bottom of the tank and the amount of water in gallons remaining in the tank after \(x\) minutes is given by $$A(x)=100\left(1-\frac{x}{5}\right)^{2}$$ (a) Calculate the average rate of change of \(A\) from 1 to 1.5 and from 2 to \(2.5 .\) Interpret your results. (b) Are the two average rates of change the same or different? Explain why.

Step-by-Step Solution

Answer

The average rates are different, -30 and -22 gallons/minute.

Step 1: Understand the Function

The given function represents how much water is left in the tank after a certain time. It is described as \( A(x) = 100\left(1 - \frac{x}{5}\right)^{2} \). This function shows that the amount of water decreases over time.

Step 2: Calculate Average Rate of Change from 1 to 1.5

To find the average rate of change, use the formula: \( \frac{A(b) - A(a)}{b - a} \). Insert \( a = 1 \) and \( b = 1.5 \):1. Calculate \( A(1) \): \( A(1) = 100\left(1 - \frac{1}{5}\right)^2 = 100\left(\frac{4}{5}\right)^2 = 64 \). 2. Calculate \( A(1.5) \): \( A(1.5) = 100\left(1 - \frac{1.5}{5}\right)^2 = 100\left(\frac{3.5}{5}\right)^2 = 49 \). 3. Find the rate: \( \frac{49 - 64}{1.5 - 1} = \frac{-15}{0.5} = -30 \). The average rate of change is -30 gallons per minute.

Step 3: Calculate Average Rate of Change from 2 to 2.5

Use the same average rate of change formula for \( a = 2 \) and \( b = 2.5 \):1. Calculate \( A(2) \): \( A(2) = 100\left(1 - \frac{2}{5}\right)^2 = 100\left(\frac{3}{5}\right)^2 = 36 \). 2. Calculate \( A(2.5) \): \( A(2.5) = 100\left(1 - \frac{2.5}{5}\right)^2 = 100\left(\frac{2.5}{5}\right)^2 = 25 \). 3. Find the rate: \( \frac{25 - 36}{2.5 - 2} = \frac{-11}{0.5} = -22 \). The average rate of change is -22 gallons per minute.

Step 4: Compare the Average Rates

The average rate of change from 1 to 1.5 is -30 gallons per minute, while from 2 to 2.5 it is -22 gallons per minute. These rates are different, indicating that the amount of water leaving the tank decreases as time progresses.

Key Concepts

Exponential Functions

An exponential function comes into play when a quantity changes at a rate proportional to its current value. In our exercise, the function describing the water level in a tank, \( A(x) = 100\left(1 - \frac{x}{5}\right)^{2} \), has components of an exponential-like form, displayed as a polynomial. Here, the term \( (1-\frac{x}{5})^{2} \) reduces the total gallons exponentially as \( x \) increases, due to squaring of the fraction. This type of function often models processes where changes occur swiftly at first, such as draining a tank quickly at the beginning, and then slowing down as time goes on. Exponential functions can represent a fast-decreasing or growing process, which is crucial in understanding how processes evolve over time. Recognizing patterns like these in real-world scenarios helps predict outcomes and make informed decisions. It's essential to have a good grasp of these functions to interpret and predict various natural and man-made phenomena effectively.

Cylindrical Tank Problem

The cylindrical tank problem involves understanding how a liquid volume changes over time when it is being drained. In our problem, the tank starts with 100 gallons of water, and the amount remaining is described by \( A(x) = 100\left(1 - \frac{x}{5}\right)^{2} \). A few things to keep in mind about cylindrical tanks:

• The water drains through the bottom, allowing gravity to aid in its release.
• The problem focuses on understanding the change in water volume over given time intervals, such as from 1 to 1.5 minutes.
• The shape and dimensions contribute to how the rate of drainage decreases as time progresses.

The significance of this problem is learning to predict how quickly or slowly a fluid will leave a container, which is incredibly useful for engineering and environmental science fields. Recognizing these changes enables better planning for situations like water management and resources allocation.

Gallons Per Minute

Gallons per minute (GPM) serves as a measure of flow rate, indicating how much water, in gallons, moves from one place to another within a minute. In our context, it's crucial for understanding the average rate of change in water level when the tank is draining. We calculated two different rates for time intervals 1 to 1.5 minutes, and 2 to 2.5 minutes. The math involved helps interpret these values:

• The calculation from 1 to 1.5 minutes resulted in a rate of -30 GPM, meaning the water level decreases by 30 gallons each minute during that interval.
• From 2 to 2.5 minutes, the rate was -22 GPM, indicating a slower decrease compared to the previous interval.

This difference in GPM highlights that the water exits more quickly at the start, and then as the level drops, the flow rate also reduces. It's like letting air out of a balloon—the initial burst is rapid, slowing as the pressure diminishes. Understanding GPM is vital in fields like plumbing, irrigation, and anywhere flow rate management becomes crucial.