Problem 23 - 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. Salem, OR (Latitude: \(44.9^{\circ} \mathrm{N}\))

Step-by-Step Solution

Answer

0.3907E

Step 1: Identify Given Values

We are given the expression for the maximum amount of light energy as \(E \sin (113.5^{\circ} + \phi)\). For Salem, OR, the latitude \(\phi\) is \(44.9^{\circ}\).

Step 2: Insert the Latitude Value

Substitute \(\phi = 44.9^{\circ}\) into the expression: \[E \sin (113.5^{\circ} + 44.9^{\circ})\].

Step 3: Calculate the Angle

Add \(113.5^{\circ}\) and \(44.9^{\circ}\): \(113.5^{\circ} + 44.9^{\circ} = 158.4^{\circ}\).

Step 4: Evaluate the Sine Function

Compute \(\sin(158.4^{\circ})\). Use a calculator to determine the value, \(\sin(158.4^{\circ}) \approx 0.3907\).

Step 5: Express in Terms of E

Multiply the sine value by \(E\): \(E \times 0.3907 = 0.3907E\).

Key Concepts

Latitude and Longitude

Latitude and longitude are the fundamental concepts used to define locations on Earth's surface. They serve as a coordinate system that allocates a unique address to every point on the planet.

• Latitude: This measures how far north or south a location is from the Equator, which is at 0°. Latitudes range from 0° at the Equator to 90° at the poles (90° N for the North Pole and 90° S for the South Pole). In the exercise, Salem, OR is located at 44.9° N, indicating it's relatively close to the northern mid-latitude.
• Longitude: This measures how far east or west a location is from the Prime Meridian, which runs through Greenwich, England, marked at 0°. Longitudes are measured up to 180° east or west from this point.

Latitude, as shown, is pivotal in the provided exercise for calculating the amount of sunlight a location like Salem, OR receives. Thus, geographical coordinates are not just for navigation but play a key role in various scientific calculations.

Sine Function

The sine function is a crucial part of trigonometry, representing the relationship between the sides of a right-angled triangle. When dealing with angles, it helps determine the vertical component relative to the hypotenuse. In simpler terms, \(\sin(\theta)\) for an angle \(\theta\) gives how much that angle is inclined vertically.

This function is represented mathematically as:

• The sine of an angle \(\theta\) is the y-coordinate of the corresponding point on a unit circle.
• For any angle \(\theta\), \(\sin(\theta)\) ranges from -1 to 1, indicating the vertical height.
• In the context of the exercise provided, \(\sin(158.4^\circ)\) calculates the factor of light energy that actually contacts the location's surface after considering the earth’s tilt and location’s latitude.

Understanding the sine function is critical as it not only helps in geometry and trigonometry but directly translates in real-world calculations like determining solar energy, waves, and oscillations.

Mathematical Calculations

Mathematical calculations form the backbone of analyzing and interpreting data in scientific exercises, especially those involving natural phenomena. In the given exercise, several arithmetic operations and trigonometric evaluations come into play.

• The initial step involves using addition to sum the angle given by the problem formula, 113.5°, with the provided latitude, 44.9°, resulting in a total angle of 158.4°.
• By applying the sine function on this angle, \(\sin(158.4^\circ)\), the relevance of this angle in determining the position of the sun relative to the location is captured.
• The sine value, obtained as approximately 0.3907 through a calculator, is then multiplied by \(E\) to express the energy in explicit terms due to its reduction factor.

These basic calculations transform abstract numerical data into useful physical information, making sense of complex natural processes simply and effectively.