Problem 68 - Algebra and Trigonometry

Question

Mixtures and Concentrations \(A 50\) -gallon barrel is filled completely with pure water. Salt water with a concentration of 0.3 Ib/gal is then pumped into the barrel, and the resulting mixture overflows at the same rate. The amount of salt in the barrel at time \(t\) is given by $$ Q(t)=15\left(1-e^{-0.04 t}\right) $$ where \(t\) is measured in minutes and \(Q(t)\) is measured in pounds. (a) How much salt is in the barrel after 5 \(\mathrm{min} ?\) (b) How much salt is in the barrel after 10 \(\mathrm{min}\) ? (c) Draw a graph of the function \(Q(t)\) (d) Use the graph in part (c) to determine the value that the amount of salt in the barrel approaches as \(t\) becomes large. Is this what you would expect?

Step-by-Step Solution

Answer

(a) \(Q(5) \approx 2.77\) lb; (b) \(Q(10) \approx 4.92\) lb; (c) Graph shows \(Q(t)\); (d) Approaches 15 lb, as expected.

Step 1: Calculate Salt Amount at 5 Minutes

To find the amount of salt in the barrel after 5 minutes, substitute \(t = 5\) into the expression for \(Q(t)\). This gives us the formula: \[Q(5) = 15(1 - e^{-0.04 \times 5})\]. Calculate this by evaluating \(e^{-0.2}\) and then solving the equation.

Step 2: Calculate Salt Amount at 10 Minutes

To determine the amount of salt in the barrel after 10 minutes, substitute \(t = 10\) into the function for \(Q(t)\): \[Q(10) = 15(1 - e^{-0.04 \times 10})\]. Calculate this by finding the value of \(e^{-0.4}\) and then computing \(Q(10)\).

Step 3: Graph the Function Q(t)

Plot the function \(Q(t) = 15(1 - e^{-0.04t})\) on a graph. Use a range of \(t\) values, starting from \(t = 0\) to a large value, for example, \(t = 50\) minutes. The x-axis should represent time \(t\) and the y-axis should represent the amount of salt \(Q(t)\).

Step 4: Determine the Limiting Value of Salt as t Increases

Observe the graph plotted in Step 3. As \(t\) becomes large, \(e^{-0.04t}\) approaches 0, and thus \(Q(t)\) approaches 15. This is consistent with the saturation level, as the maximum concentration of salt that can be maintained in the barrel is 0.3 lb/gal and the total volume is 50 gallons.

Key Concepts

Exponential Decay

Exponential decay is a mathematical concept describing how a quantity decreases rapidly at first and then more slowly over time. In our problem, the function for the amount of salt in the barrel, \(Q(t) = 15(1 - e^{-0.04t})\), incorporates exponential decay. The term \(e^{-0.04t}\) signifies this decay process. As time passes, the value of the exponential term gets smaller, meaning the effect of time on reducing the difference between maximum potential salt content and current salt content diminishes.Exponential decay is common in processes where changes are proportionally related to the current state of a system, such as radioactive decay, cooling, and dissolving substances in water. Understanding this behavior helps predict when a system will approach stability or equilibrium. The function \(Q(t)\) approaches an upper limit, which, in this case, is the saturation point of salt in the barrel.

Salt Concentration

Salt concentration refers to the amount of salt within a given volume of a solution. In our exercise, salt water with a concentration of 0.3 lb/gal is continuously added to the barrel, which initially contains pure water. The concentration of salt affects how quickly the solution in the barrel reaches saturation.Here, the function \(Q(t)\) models how the salt amount changes over time in response to the inflow of the salt solution. Key points about salt concentration include:- It determines the saturation level or the maximum amount of salt the solution can dissolve before the concentration levels off.- When the concentration in the barrel reaches equilibrium with the inflow rate, the increase of salt concentration stops.This practical understanding of salt concentration is vital in applications ranging from food processing to chemical manufacturing.

Time-Dependent Function

A time-dependent function is a mathematical representation showing how a certain variable or quantity changes over time. In this scenario, \(Q(t)\), which represents the amount of salt in the barrel, is a time-dependent function. The equation involves the variable \(t\), representing time in minutes, showcasing how the salt content changes as time progresses.Functions that are dependent on time are widely applicable across different fields:- In physics, they can represent motion or decay.- In finance, they can model growth over time.In this problem,

• \(Q(t)\) starts at 0 when the barrel has pure water.
• As \(t\) increases, \(Q(t)\) rises, approaching a maximum of 15 pounds of salt.

Such functions provide valuable insights into developing patterns over time, allowing us to predict the future state of systems from past data.

Graphing Functions

Graphing functions is a tool that allows visualization of how variables in an equation relate. For the function \(Q(t) = 15(1 - e^{-0.04t})\), graphing shows us the growth trend of the salt amount in the barrel over time.In our example:- The x-axis represents time \(t\).- The y-axis shows the quantity of salt \(Q(t)\).As you plot the graph, ideally from \(t = 0\) to a larger value like 50 minutes:

• The graph initially steepens, indicating a rapid increase in salt content.
• Then it levels off, showing saturation.

Graphing functions is crucial in mathematics and science as it provides a clear visual counterpart to numerical data, supporting deeper understanding of the behavior and relationships in mathematical expressions. It also helps identify limits, such as the maximum salt content in our example, without lengthy calculations.