Problem 44 - Applied Calculus

Question

For a new type of biofuel, scientists estimate that it takes \(A=f(g)\) gallons of gasoline to produce the raw materials to generate \(g\) gallons of biofuel. Assume the biofuel is equal in efficiency to the gasoline. (a) At a certain level of production, we have \(d A / d g=\) 1.3. Interpret this in practical terms. Is this level of production sustainable? Explain. (b) Repeat part (a) if \(d A / d g=0.2\)

Step-by-Step Solution

Answer

At 1.3, biofuel production is unsustainable; at 0.2, it is sustainable.

Step 1: Understanding the Derivative's Interpretation

The derivative \( \frac{dA}{dg} \) represents the rate of change of gasoline used (\( A \)) with respect to biofuel produced (\( g \)). A value of 1.3 means that for each additional gallon of biofuel produced, 1.3 gallons of gasoline are consumed. In practical terms, this means that producing more biofuel than gasoline is consumed is not sustainable, as it consumes more gasoline than the biofuel generated.

Step 2: Evaluating Sustainability for 1.3

For \( \frac{dA}{dg} = 1.3 \), producing 1 gallon of biofuel requires using 1.3 gallons of gasoline. Since more gasoline is used than biofuel produced, this process is unsustainable as it leads to a net loss of fuel resources.

Step 3: Repeating the Analysis for 0.2

Now, analyze the case where \( \frac{dA}{dg} = 0.2 \). Here, for every gallon of biofuel produced, only 0.2 gallons of gasoline are needed. This means creating biofuel results in a net gain of 0.8 gallons, suggesting a sustainable process since less gasoline is consumed than biofuel is generated.

Step 4: Evaluating Sustainability for 0.2

For \( \frac{dA}{dg} = 0.2 \), the production process is sustainable because it requires less gasoline than the equivalent energy produced as biofuel, indicating a net gain of fuel.

Key Concepts

Sustainability Analysis

When analyzing the sustainability of biofuel production, it is essential to evaluate whether the resources consumed are outweighed by the benefits gained. Sustainability, in this context, involves producing energy in a way that does not deplete natural resources faster than they can be replenished.
This is crucial because if biofuel production consumes more gasoline than it creates as a replacement, it defeats the purpose of moving towards renewable energy sources. In our exercise, when the rate at which gasoline is consumed (\(\dfrac{dA}{dg}\) is 1.3, this level of production is unsustainable as it requires more gasoline to produce biofuel.

• This leads to a net loss of energy, with more non-renewable fuel being used rather than replaced.
• Understanding these variables helps us make informed decisions about scaling production in an environmentally responsible way.

However, when this rate is reduced to 0.2, the process becomes sustainable as only a fraction of gasoline is consumed compared to the generated biofuel.

• This results in an overall gain, effectively reducing reliance on fossil fuels.
• Such practices are critical to reducing carbon footprints and achieving long-term ecological balance.

Biofuel Production

Biofuel production is an alternative energy source aimed at reducing dependence on fossil fuels. This process involves converting organic materials, such as plant material or animal waste, into usable energy. The benefits of biofuel include reduced greenhouse gas emissions and the potential for renewable energy.
However, its sustainability largely depends on the efficiency of the production process. A crucial factor is how much non-renewable energy (like gasoline) is used in producing biofuel.

• If the process requires more energy input than output, it may not be an effective alternative.
• Understanding the derivative of the rate of change helps gauge production efficiency.

In the exercise, understanding the value of \(\dfrac{dA}{dg} = 1.3\) highlights the inefficiency of the system since more gasoline is required than biofuel produced. Conversely, a rate of 0.2 indicates a sustainable production method, saving energy and reducing carbon emissions.

• Efficient biofuel production not only provides an eco-friendly energy source but also promotes energy security and economic stability.

Rate of Change

The rate of change is a vital concept in understanding how one quantity varies concerning another. In our case, it provides insight into how much gasoline is required as biofuel production increases. Mathematically represented by \(\dfrac{dA}{dg}\), it is the derivative illustrating a direct link between the quantity of gasoline used and the biofuel produced.
Calculating the rate of change helps determine resource efficiency. When \(\dfrac{dA}{dg} = 1.3\), it shows inefficiency as more gasoline is spent than biofuel obtained.

• This implies a negative impact environmentally and economically and suggests reevaluation of the production practices.

Conversely, a rate of 0.2 illustrates a positive shift where less gasoline is needed, suggesting an efficient and environmentally friendly operation.

• Understanding such metrics assists in improving production strategies and ensures sustainable development goals.
• Measuring and interpreting these rates empower industries to adopt smarter, resource-saving methods.

Proper analysis of the rate of change is central to ensuring sustainable expansion of renewable energy options.