Problem 91 - Calculus and its applications

Question

A factory is polluting a lake in such a way that the rate of pollutants entering the lake after \(t\) months is $$N^{\prime}(t)=280 t^{3 / 2}$$ where \(N(t)\) is the total number of pounds of pollutants in the lake after \(t\) months. a) How many pounds of pollutants enter the lake in 16 months? b) An environmental board tells the factory that it must begin cleanup procedures after \(50,000 \mathrm{lb}\) of pollutants have entered the lake. After what length of time will this occur?

Step-by-Step Solution

Answer

a) 114,688 pounds enter in 16 months. b) 50,000 lb is reached after approximately 7.34 months.

Step 1: Define the Problem

We are given the rate of pollutants entering a lake as a function of time: \(N'(t) = 280t^{3/2}\). We need to find the total amount of pollutants entering the lake after 16 months and the time after which 50,000 lb of pollutants are present.

Step 2: Integrate to Find the Total Pollutants

To find the total amount of pollutants, we need to integrate the rate function \(N'(t)\). Integrating \(N'(t) = 280t^{3/2}\) gives \[ N(t) = \int 280t^{3/2} \, dt = 280 \cdot \frac{2}{5} t^{5/2} + C = 112 t^{5/2} + C. \] We can assume \(C = 0\) if we start counting from \(t = 0\).

Step 3: Calculate N(16)

Substitute \(t = 16\) into our integrated function to find the total pollutants after 16 months. \[ N(16) = 112 \cdot (16)^{5/2}. \] First, calculate \((16)^{5/2}\) which is \((16)^{2.5} = (2^4)^{2.5} = 2^{10} = 1024\). Then, \(N(16) = 112 \times 1024 = 114,688\) pounds.

Step 4: Solve for Time to Reach 50,000 lb

Set the equation \(N(t) = 112t^{5/2} = 50,000\) to find time \(t\). Solve for \(t\): \[ t^{5/2} = \frac{50,000}{112}. \] First, compute \(\frac{50,000}{112} \approx 446.43\). Now solve \((t^{5/2} = 446.43)\). Take both sides to the power of \(2/5\): \( t = (446.43)^{2/5}.\) Calculating \((446.43)^{2/5}\), we find \(t \approx 7.34\) months.

Key Concepts

Integration

Integration is a fundamental concept in calculus that helps us find quantities like area, volume, and in this case, the accumulation of pollutants. It's the opposite of differentiation.
In our problem, the rate at which pollutants enter the lake is expressed by a derivative function, \( N'(t) = 280t^{3/2} \). To determine the total pollutants at a certain time \( t \), we must integrate this function.

• Integration reverses differentiation by summing up infinitely small quantities to find a total.
• In this context, it allows us to compute the accumulated pollutants over a period.
• The key to integration is finding an antiderivative that represents the accumulated value up to time \( t \).

In this scenario, the integration of the rate of pollutants function gives us the total quantity of pollutants, represented as:
\[ N(t) = \int 280t^{3/2} \, dt = 112t^{5/2} + C. \]Assuming no pollutants at time zero means setting \( C = 0 \). Thus, integration helps us track the total pollution over time.

Pollution Modeling

Pollution modeling is critical in understanding and managing the environmental impact of industrial activities. It involves predicting how pollutants spread and accumulate over time. This helps stakeholders make informed decisions.
In the given problem, pollution modeling is used to determine how the pollutants introduced by the factory affect the lake over time.

• The rate of pollutants entering the lake is modeled by a mathematical function in terms of time \( t \).
• We use this model to calculate how much pollution exists at any given moment by integrating the rate function.
• This is essential for setting up regulations such as the maximum allowable pollution levels.

In our case, we evaluate \( N(16) \) to find how many pounds of pollutants enter the lake over 16 months. Then, the model also helps compute the time when the pollution reaches a specified threshold, such as 50,000 pounds.

Environmental Mathematics

Environmental mathematics applies mathematical techniques to solve environmental problems. This field encompasses a variety of methods to predict, analyze, and solve issues concerning the environment, like pollution.
The industrial pollution problem is an example where mathematics serves as a valuable tool for environmental management. The steps taken in this problem demonstrate how mathematical formulations lead to practical solutions.

• We use differentiation and integration, key calculus methods, to understand and describe environmental processes.
• Mathematical models like our pollution rate function quantify and address pollution scenarios.
• This numeric modeling helps with planning environmental protection strategies and making data-driven decisions.

Through mathematical modeling, stakeholders can predict when certain thresholds are met, paving the way for timely and effective interventions to protect and preserve environmental health.