Problem 95 - College Algebra with Modeling and Visualization

Question

The U.S. consumption of energy from 1950 to 1980 can be modeled by \(f(x)=-0.00113 x^{3}+0.0408 x^{2}-0.0432 x+7.66\) where \(x=0\) corresponds to 1950 and \(x=30\) to 1980 Consumption is measured in quadrillion Btu. (Source: Department of Energy.) (a) Evaluate \(f(5)\) and interpret the result. (b) Graph \(f\) in \([0,30,5]\) by \([6,16,1]\). Describe the cnergy usage during this time period. (c) Approximate the local maximum and interpret it.

Step-by-Step Solution

Answer

(a) 8.32 quadrillion Btu in 1955. (b) Energy usage generally increased, peaked, then declined. (c) The local maximum represents peak energy usage over these years.

Step 1: Define the function

The function provided is a cubic polynomial defined as: \[ f(x) = -0.00113 x^{3} + 0.0408 x^{2} - 0.0432 x + 7.66 \] Here, \( x \) represents the number of years since 1950, and \( f(x) \) gives the energy consumption in quadrillion Btu.

Step 2: Evaluate f(5)

To find the energy consumption in 1955 (5 years after 1950), substitute \( x = 5 \) into the function: \[ f(5) = -0.00113(5)^3 + 0.0408(5)^2 - 0.0432(5) + 7.66 \] Calculate each term: - \(-0.00113(5^3) = -0.00113(125) = -0.14125\) - \(0.0408(25) = 1.02\) - \(0.0432(5) = 0.216\) Thus, \[ f(5) = -0.14125 + 1.02 - 0.216 + 7.66 = 8.32275 \] Therefore, in 1955, the energy consumption was approximately 8.32 quadrillion Btu.

Step 3: Graph the function over [0, 30]

Graph the function \( f(x) \) over the interval \([0, 30]\) with an interval of 5. The range for graphing is set from 6 to 16 with steps of 1. This graphical representation will provide a visualization of how energy consumption changed from 1950 to 1980.

Step 4: Describe energy usage trend

Upon observing the graph, note the trend of the curve representing energy consumption. Generally, it may show an initial increase, peak at some point, and then possibly decrease as per the cubic nature of the polynomial. Describe these trends specific to periods such as early, mid, and late years between 1950 and 1980.

Step 5: Find local maximum

To approximate the local maximum, identify periods where the derivative \( f'(x) \) equals zero. The derivative of our function is: \[ f'(x) = -0.00113(3)x^2 + 0.0408(2)x - 0.0432 \] This simplifies to: \[ f'(x) = -0.00339x^2 + 0.0816x - 0.0432 \] Solve \( f'(x) = 0 \) to find critical points. However, numerical methods like a graphing calculator or software might be more practical for determining where the turning point occurs within the specified period.

Step 6: Interpret the local maximum

Once the local maximum has been approximated using graphing or calculation software, interpret this point in terms of its year and the corresponding energy consumption. This maximum symbolizes the peak energy use within the studied period from 1950 to 1980.

Key Concepts

Energy Consumption Modeling

Energy consumption modeling involves creating mathematical representations of energy use over time. In this exercise, a cubic polynomial function is used to model energy consumption in the U.S. between 1950 and 1980. The function \(f(x) = -0.00113x^3 + 0.0408x^2 - 0.0432x + 7.66\) provides valuable insights into how energy consumption evolved during this period. Each term in the polynomial reflects different influences on consumption:

• The cubic term \(-0.00113x^3\) might represent complex factors affecting energy usage trends such as technological advancements or economic conditions.
• The quadratic term \(0.0408x^2\) captures acceleration or deceleration over time.
• The linear term \(-0.0432x\) directly corresponds to predictable linear changes in usage.
• The constant term \(7.66\) sets the base level of consumption at the start of the period.

This model helps us understand past patterns and can inform future predictions, making it an essential tool in analyzing energy policies and planning.

Calculus and Derivatives

In calculus, derivatives are fundamental as they provide information about the rates of change. For modeling energy consumption, the derivative of the cubic polynomial function \(f(x)\) helps identify behaviors like growth rates and turning points. The derivative is calculated as:
\[ f'(x) = -0.00339x^2 + 0.0816x - 0.0432 \]
Here's what this derivative tells us:

• If \(f'(x) > 0\), the energy consumption is increasing. This indicates when the rate of consumption is rising.
• If \(f'(x) < 0\), the consumption is decreasing, showing a decline in the rate of consumption.
• Where \(f'(x) = 0\), critical points or possible local maxima/minima may exist. In practical terms, it might highlight the year where consumption reaches a peak before declining.

Finding these critical points requires solving \(f'(x) = 0\), which can be challenging with polynomial equations. Often, numerical techniques or graphing technology help make these complex calculations more manageable.

Graph Interpretation

Graph interpretation is crucial as it provides a visual representation of numerical data, offering insights at a glance. When graphing the function \(f(x)\), you are essentially plotting energy consumption over time. The range of \(x\) spans from 0 to 30, corresponding to the years from 1950 to 1980. By graphing this:

• You can see the initial increase in energy consumption, which might symbolize post-war industrial growth and economic expansion.
• The graph could show a peak, indicating maximum energy use at a certain point, which aligns with a critical point found through calculus.
• A subsequent decrease may reflect changes such as energy efficiency measures or economic downturns.

Visually identifying trends helps contextualize the data within real-world events, enhancing the understanding of energy use patterns during the period studied. By using graphs, complex data becomes approachable, allowing for more informed decisions in energy management.