Problem 26 - Algebra 2

Question

On December \(22,\) the maximum amount of light energy that falls on a square foot of ground at a certain location is given by \(E \sin \left(113.5^{\circ}+\phi\right),\) where \(\phi\) is the latitude of the location. Find the amount of light energy, in terms of \(E,\) for each location. San Diego, CA (Latitude \(32.7^{\circ} \mathrm{N}\))

Step-by-Step Solution

Answer

The light energy in San Diego is \(E \sin(146.2^{\circ})\).

Step 1: Identify the Given Information

The problem provides the maximum light energy formula as \(E \sin \left(113.5^{\circ}+\phi\right)\). We are also given that the latitude \(\phi\) for San Diego is \(32.7^{\circ}\).

Step 2: Substitute the Latitude into the Formula

Replace \(\phi\) with \(32.7^{\circ}\) in the given formula for light energy.\[ E \sin \left(113.5^{\circ} + 32.7^{\circ}\right) \]

Step 3: Simplify the Angle

Add the angles inside the sine function:\[ 113.5^{\circ} + 32.7^{\circ} = 146.2^{\circ} \]

Step 4: Calculate the Sine of the Angle

Compute the sine of the angle:\[ \sin(146.2^{\circ}) \]

Step 5: Express the Result in Terms of E

Finally, express the light energy in terms of \(E\):\[ E \sin(146.2^{\circ}) \]

Key Concepts

Latitude

Latitude is a geographical term used to specify the location of a place on Earth's surface. It is the angular distance measured in degrees, north or south from the equator. The equator is marked as 0 degrees latitude, and the poles as 90 degrees north or south.

Latitude plays a significant role in determining climate and the amount of sunlight an area receives. Places closer to the equator receive more direct sunlight year-round, while those near the poles encounter more significant variations in day length and sun angle, depending on the season.

For example, San Diego, CA, has a latitude of 32.7 degrees North. This value indicates that San Diego is situated north of the equator and receives a specific amount of sunlight based on its position on the Earth’s surface. Understanding latitude is crucial for calculating ecological elements such as solar energy adsorption as seen in the given exercise.

Sine function

The sine function is an essential trigonometric function that relates an angle of a right triangle to the ratio of the length of the opposite side to its hypotenuse. In mathematics, the sine of an angle \( \theta \) is commonly written as \( \sin(\theta) \).

In the context of the problem, the sine function is used to determine the proportion of maximum light energy received at different angles. The formula \( E \sin(113.5^{\circ} + \phi) \) integrates the angle of 113.5 degrees with the latitude \( \phi \) to calculate light energy.

• Sine functions have values ranging from -1 to 1, representing the amplitude of waveforms.
• The sign (positive or negative) of the sine function depends on the angle, influencing the energy calculation.

By calculating \( \sin(146.2^{\circ}) \), which results from the sum of 113.5 degrees and San Diego’s latitude (32.7 degrees), you can determine the fraction of the maximum light energy \(E\) that the location receives.

Light energy calculation

Calculating light energy involves understanding and applying mathematical formulas that take into account the position of the Earth relative to the sun. The formula provided, \( E \sin(113.5^{\circ} + \phi) \), is used to find out how much light energy, expressed as \(E\), falls on a surface at different times of the year based on the Earth's tilt and rotation.

This formula includes:

• \(E\): a constant representing the maximum energy the area can receive
• Sine function: Calculating the angle’s sine that combines a fixed solar position of \(113.5^{\circ}\) plus the latitude

This makes it possible to examine how sunlight intensity changes with the season or based on geographical position. The multiplication of \(E\) and the sine function calculates the effective light energy.

In San Diego, the computation of \(E \sin(146.2^{\circ})\) shows that the location does not receive the absolute maximum energy during this specific time, which is consistent with it being winter in the northern hemisphere. Understanding these calculations is crucial for fields like solar energy, where knowing the potential energy available at a given location is vital.