Problem 91 - Algebra A Combined Function

Question

The polynomial \(-20 x^{2}+156 x+14,437\) represents he electricity generated (in gigawatts) by geothermal sources in the United States during \(2002-2007\). The polynomial \(894 x^{2}-90 x+10,939\) represents the electricity generated (in gigawatts) by wind power in the United States during \(2002-2007\). In both polynomials, \(x\) represents the number of years after 2002 . Find a polynomial for the total electricity generated by both geothermal and wind power during 2002-2007. (Source: Based on information from the Energy Information Administration)

Step-by-Step Solution

Answer

The total polynomial is \(874x^2 + 66x + 25,376\).

Step 1: Identify the Given Polynomials

We are given two polynomials that represent the electricity generated by geothermal and wind power sources. The polynomial for geothermal is \(-20x^2 + 156x + 14,437\), and the polynomial for wind power is \(894x^2 - 90x + 10,939\).

Step 2: Set Up the Addition Operation

To find a polynomial for the total electricity generated by both sources, we need to add the two given polynomials together. This involves adding corresponding coefficients of \(x^2\), \(x\), and the constant terms.

Step 3: Add the Coefficients of \(x^2\) Terms

Add the coefficients of the \(x^2\) terms from both polynomials: \(-20 + 894 = 874\). Thus, the coefficient for the \(x^2\) term in the total polynomial is \(874\).

Step 4: Add the Coefficients of \(x\) Terms

Add the coefficients of the \(x\) terms from both polynomials: \(156 - 90 = 66\). Thus, the coefficient for the \(x\) term in the total polynomial is \(66\).

Step 5: Add the Constant Terms

Add the constant terms from both polynomials: \(14,437 + 10,939 = 25,376\). Thus, the constant term in the total polynomial is \(25,376\).

Step 6: Form the Total Polynomial

Combine the results from the previous steps to form the total polynomial: \(874x^2 + 66x + 25,376\).

Key Concepts

Geothermal Energy

Geothermal energy is a renewable energy source that originates from the heat stored beneath the Earth's surface. This energy is harnessed by tapping into underground reservoirs of steam and hot water, which can be used to drive turbines and generate electricity. The key advantage of geothermal energy is its consistency; it can provide a continuous power supply regardless of weather conditions compared to solar or wind energy.

The production of electricity via geothermal energy in the United States is represented by the polynomial \(-20x^2 + 156x + 14,437\), where \(x\) indicates the number of years after 2002. Here, the coefficients and constant provide insights into how annual generation is expected to change over time. For example:

• The \(-20x^2\) term suggests a potential decrease in growth rate over time, possibly due to depletion of easily accessible resources or increased operational challenges.
• The \(156x\) indicates an initial increase in geothermal electricity generation capacity as facilities may have expanded or technological advancements may have improved efficiency.
• The constant term \(14,437\) represents the baseline electricity generation capacity from geothermal sources at the start of the period in 2002.

Understanding these factors is crucial as it helps assess how geothermal energy contributes to the overall energy mix and planning for future capacity expansions or upgrades.

Wind Power

Wind power leverages the kinetic energy of moving air by using wind turbines to convert this energy into electricity. It's a clean and abundant form of energy that has become a significant player in the renewable energy landscape. In the U.S., the generation of wind power during the period from 2002 to 2007 is described by the polynomial \(894x^2 - 90x + 10,939\).

By examining the components of this polynomial:

• The \(894x^2\) term reflects the rapid expansion and investment in wind energy infrastructure. This substantial coefficient highlights aggressive growth in capacity likely due to advances in turbine technology and favorable policies.
• The \(-90x\) coefficient is indicative of possible teething issues or operational setbacks that were overcome over time.
• The constant \(10,939\) captures the initial baseline of wind power generation as of 2002.

The polynomial sums up a period of significant growth in wind power, highlighting the technological and infrastructural progress. Understanding this growth can guide future developments in renewable energy strategies and policy-making.

Electricity Generation

Electricity generation is the process of producing electric power from various energy sources. In context, geothermal and wind power are among the diverse portfolio of renewable energy sources contributing to sustainable electricity generation.

Adding the polynomials representing geothermal and wind power provides a comprehensive view of their combined impact on electricity generation from 2002 to 2007. The polynomial sum is: \(874x^2 + 66x + 25,376\). This equation represents the collective contribution of energy generated from both geothermal and wind sources over the given period.

Breaking down the combined polynomial:

• The \(874x^2\) term signifies a robust trend in capacity expansion due to both geothermal and wind energy projects.
• The \(66x\) term indicates an overall positive and steady growth in electricity generation capabilities, suggesting strengthening infrastructure efficiency.
• The \(25,376\) constant illustrates the initial baseline generation capacity combining both renewable sources as of 2002.

This analysis reveals how both energy sources jointly contribute to meeting electricity demand through renewable means, reinforcing their importance in a balanced energy strategy that aims to reduce greenhouse gas emissions and reliance on fossil fuels.