Problem 40 - Calculus

Question

The power output, \(P\), of a solar panel varies with the position of the sun. Let \(P=10 \sin \theta\) watts, where \(\theta\) is the angle between the sun's rays and the panel, \(0 \leq \theta \leq \pi\) On a typical summer day in Ann Arbor, Michigan, the sun rises at 6 am and sets at \(8 \mathrm{pm}\) and the angle is \(\theta=\pi t / 14,\) where \(t\) is time in hours since 6 am and \(0 \leq t \leq 14\) (a) Write a formula for a function, \(f(t),\) giving the power output of the solar panel (in watts) \(t\) hours after 6 am on a typical summer day in Ann Arbor. (b) Graph the function \(f(t)\) in part (a) for \(0 \leq t \leq 14\) (c) At what time is the power output greatest? What is the power output at this time? (d) On a typical winter day in Ann Arbor, the sun rises at 8 am and sets at 5 pm. Write a formula for a function, \(g(t),\) giving the power output of the solar panel (in watts) \(t\) hours after 8 am on a typical winter day.

Step-by-Step Solution

Answer

(a) \( f(t) = 10 \sin\left(\frac{\pi t}{14}\right) \). (c) Max output at 1 pm is 10 watts.

Step 1: Define the function for the summer

The given power output formula is \( P = 10 \sin \theta \). We have \( \theta = \frac{\pi t}{14} \) for a summer day. Substitute \( \theta \) into the power output equation. This yields the summer power function as \( f(t) = 10 \sin\left(\frac{\pi t}{14}\right) \).

Step 2: Graph the function

To graph \( f(t) = 10 \sin\left(\frac{\pi t}{14}\right) \), create a table of values for \( t \) from 0 to 14, and calculate corresponding \( f(t) \). Plot these points and connect them to form the curve from 6 am to 8 pm, where \( t = 0 \) corresponds to 6 am.

Step 3: Determine maximum power output for summer

The power function \( f(t) = 10 \sin\left(\frac{\pi t}{14}\right) \) is maximized when \( \sin\left(\frac{\pi t}{14}\right) = 1 \). This occurs at \( \frac{\pi t}{14} = \frac{\pi}{2} \). Solve for \( t \) to find \( t = 7 \). Thus, the maximum power output is 10 watts at 1 pm (since 6 am + 7 hours = 1 pm).

Step 4: Define the function for the winter

For winter with sun rising at 8 am, \( \theta = \frac{\pi t}{9} \) since the sun sets at 5 pm, covering 9 hours. Substitute into the power equation: \( g(t) = 10 \sin\left(\frac{\pi t}{9}\right) \). This describes the power output during winter.

Key Concepts

Solar Energy

Solar energy is the power harnessed from the sun's rays. Solar panels capture this energy and convert it into electricity.
They are a popular renewable energy source with minimal environmental impact.
Understanding how solar panels work can help maximize their efficiency. - **Energy Conversion:** Solar panels convert sunlight into electrical energy using photovoltaic cells. - **Dependence on Sun's Position:** The ability of a solar panel to capture energy greatly depends on the angle of the sunlight striking it. The more directly the sun's rays hit the panel, the more energy it can capture. - **Peak Hours:** Usually, solar panels generate the most power when the sun is highest in the sky, which typically occurs around noon. In the given exercise, the efficiency of solar panels in different seasons is explored through trigonometric functions, demonstrating how the angle of sunlight affects power output.

Sine Function

The sine function is a fundamental concept in trigonometry, used to model periodic phenomena like the fluctuations in solar power output.
This exercise showcases the sine function to describe how solar energy varies over time.- **Mathematical Representation:** The sine function is defined as \( y = ext{opposite} / ext{hypotenuse} \), appearing as a wave when graphed.- **Function Behavior:** As an angle changes from 0 to \( 2\pi \), the sine value varies from 0 to 1, back to 0, to -1, and finally back to 0.- **Application:** In this case, \( P = 10 \sin \frac{\pi t}{14} \), this formula shows how the power output oscillates like a sine wave over the daylight hours.This sinusoidal behavior helps predict when the solar panel will deliver maximum energy output, which is when \( \sin \theta = 1 \). This results in maximum power, showing precisely how mathematics underpins solar energy management.

Graphing Functions

Graphing functions is an essential skill in understanding relationships between variables.
It allows us to visually interpret how the power output from a solar panel changes throughout the day.- **Creating a Graph:** Begin by calculating values of the function at various points. For instance, to graph \( f(t) = 10\sin\left(\frac{\pi t}{14}\right) \), determine values of \( t\) from 0 to 14, then compute \( f(t) \).- **Plotting Points:** Each computed value of \( f(t) \) corresponds to a point on the graph.- **Forming the Curve:** Connect these points smoothly to represent the periodic nature of the sine function.Graphing makes it easier to determine peak power output visually. In this exercise, you can readily see when power is at its maximum and minimum just by examining the curve. This aids in planning energy usage and enhances understanding of how solar panels operate at different times of the day.