Problem 10 - Applied Calculus

Question

Solar photovoltaic (PV) cells are the world's fastest growing energy source. In year \(t\) since \(2007,\) PV cells were manufactured worldwide at a rate of \(S=3.7 e^{0.61 t}\) gigawatts per year. \(^{9}\) Estimate the total solar energy generating capacity of the PV cells manufactured between 2007 and 2011.

Step-by-Step Solution

Answer

The total capacity is approximately 63.61 gigawatts.

Step 1: Understand the Problem

We need to find the total solar energy generating capacity of PV cells manufactured between 2007 and 2011. The function given, \( S = 3.7 e^{0.61 t} \), describes the rate of PV cell manufacturing in gigawatts per year, where \( t \) is the number of years since 2007.

Step 2: Set the Bounds for Integration

Since we are looking for the capacity manufactured between 2007 and 2011, the time \( t \) ranges from \( t = 0 \) (for 2007) to \( t = 4 \) (for 2011). These bounds will be used in our integration.

Step 3: Set up the Integral

To find the total energy capacity manufactured during this period, integrate the rate function \( S(t) = 3.7 e^{0.61 t} \) with respect to \( t \) from 0 to 4:\[\int_{0}^{4} 3.7 e^{0.61 t} \, dt.\]

Step 4: Calculate the Integral

Calculate the integral \[\int 3.7 e^{0.61 t} \, dt = \frac{3.7}{0.61} e^{0.61 t} + C,\]where \( C \) is the constant of integration. Evaluating this expression from 0 to 4 gives:\[\left[ \frac{3.7}{0.61} e^{0.61 \times 4} \right] - \left[ \frac{3.7}{0.61} e^{0} \right].\]

Step 5: Compute the Values

First, calculate \( \frac{3.7}{0.61} \approx 6.0656 \). Substitute into the evaluated integral:\[6.0656 \times e^{2.44} - 6.0656 \times e^{0}.\]Compute \( e^{2.44} \approx 11.485 \), thus:\[6.0656 \times 11.485 - 6.0656 \times 1 \approx 69.68 - 6.0656 \approx 63.61.\]

Step 6: State the Total Estimated Capacity

The total solar energy generating capacity of the PV cells manufactured between 2007 and 2011 is approximately 63.61 gigawatts.

Key Concepts

Photovoltaic Cells

Photovoltaic (PV) cells are a revolutionary technology transforming sunlight directly into electricity. At the heart of a PV cell is a semiconductor material, commonly silicon, that absorbs photons from sunlight. When photons hit the semiconductor, they energize electrons, knocking them loose and allowing them to flow as electricity. This electron flow is harnessed as a current which can be used to power electrical devices or stored for future use.
PV cells are used in a multitude of applications, from small calculators to large solar farms. They offer a clean and renewable energy source, helping to reduce the carbon footprint of energy production. This has contributed to their status as one of the fastest-growing energy sources globally.
Key advantages of PV cells include:

• Non-polluting: They generate electricity without harmful emissions.
• Scalability: They can be used in small scales for residential use or large scales for industrial electricity generation.
• Low maintenance: PV systems generally require minimal maintenance once installed.

The worldwide growth in the manufacturing of PV cells reflects increasing environmental awareness and technological advancement, making solar energy more accessible and affordable.

Integration in Calculus

Integration in calculus is an essential tool that allows us to calculate areas under curves, among many other applications. In simplest terms, integration is the mathematical process of summing an infinite number of infinitesimally small quantities to find total values, such as areas, volumes, or in the case of solar energy, total capacity.
In the given problem, we used integration to find the total energy generating capacity of solar photovoltaic cells produced between 2007 and 2011. We did this by integrating the function that represents the rate of cell production. The function provided, \[ S(t) = 3.7 e^{0.61 t} \], gives the production rate in gigawatts per year. By integrating this function over the time interval from 0 to 4, representing the years 2007 to 2011:
\[ \int_{0}^{4} 3.7 e^{0.61 t} \, dt \].
This integral calculates the total area under the curve of the production rate function, translating directly to the total amount of solar capacity produced over that period. Integration thus serves as a powerful method to compute cumulative quantities based on known rate functions.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases with passing time, creating a curve that is steep and upward sloping on a graph. This concept is key in understanding the rapid advancement of solar technology, including the production of photovoltaic cells.
In the exercise example, the rate of producing PV cells follows an exponential growth model represented by the expression \[ S(t) = 3.7 e^{0.61 t} \]. This formula indicates that production grows exponentially due to the base of the exponential function \( e \) being positive. Exponential growth occurs in contexts where growth accelerates over time, multiplying by a consistent factor over equivalent time periods.
It's noteworthy:

• Real-world applicability: Exponential growth is evident in populations, investments, and technologies, where results over time multiply rapidly.
• Potential for high impact: In the context of solar energy, exponential growth aids in achieving substantial energy independence in a relatively short time frame.
• Forecasting trends: By understanding exponential trends, companies and policymakers can better plan for the future energy needs and technology advancements.

Exponential growth plays a critical role in moving forward with renewable energy as it showcases how swiftly improvements and increases in production capacity are making solar energy a more dominant player in the global energy market.