Problem 39 - Algebra 2

Question

UCHTING For Exercises 39 and 40 , use the following information. The amount of light that a source provides to a surface is called the illuminance. The illuminance \(E\) in foot candles on a surface is related to the distance \(R\) in feet from the light source. The formula sec \(\theta=\frac{1}{E R^{2}}\) where \(I\) is the intensity of the light source measured in candles and \(\theta\) is the angle between the light beam and a line perpendicular to the surface, can be used in situations in which lighting is important. Solve the formula in terms of \(E\)

Step-by-Step Solution

Answer

\( E = \frac{\cos \theta}{R^{2}} \)

Step 1: Write Down the Given Formula

The given formula is \( \sec \theta = \frac{1}{E R^{2}} \). We have to solve this formula in terms of \( E \).

Step 2: Isolate the Term Involving E

Start simplifying by taking the reciprocal of both sides of the equation. This gives us \( E R^{2} = \frac{1}{\sec \theta} \).

Step 3: Express in Terms of Cosine

Recall the identity \( \sec \theta = \frac{1}{\cos \theta} \). Substitute this into the equation: \( E R^{2} = \cos \theta \).

Step 4: Solve for E

Divide both sides of the equation by \( R^{2} \) to isolate \( E \): \( E = \frac{\cos \theta}{R^{2}} \).

Key Concepts

Light intensity

Light intensity refers to the strength of light emitted from a source. It is quantified in terms of 'candles' or 'candelas.' When a light source emits light, it shines in all directions. The light intensity helps determine how much light is being emitted overall.

The stronger the intensity, the more illuminance it can provide to a surface. This is especially important in lighting design and security. For instance, rooms with studying or working areas require higher light intensity for better visibility. Understanding this concept helps you realize how important it is to choose the correct lighting for different environments.

• Higher intensity means more light is emitted.
• Measured in candles or candelas.
• Plays a crucial role in determining illuminance.

Distance and light

The distance between a light source and a surface, often denoted as \( R \), inversely affects the illuminance that surface receives. As the distance \( R \) increases, the light becomes more spread out and thus decreases in intensity per unit area. Hence, the illuminance on the surface reduces.

Mathematically, illuminance \( E \) is proportional to the inverse square of the distance, which is expressed in the relation \( E \propto \frac{1}{R^2} \). This inverse square law is crucial when planning lighting for large areas. For designers or architects, understanding this relationship ensures that adequate lighting can be provided, even as the distance varies.

• Illuminance decreases as distance increases.
• Follows the inverse square law \( (E \propto \frac{1}{R^2}) \).
• Essential for efficient lighting design.

Trigonometric identities

Trigonometric identities are mathematical tools that help simplify and solve equations involving angles. One of the identities used in the light formula is \( \sec \theta = \frac{1}{\cos \theta} \). This expression transforms the original equation, making it easier to work with. When the secant function is expressed in terms of cosine, it enables us to rewrite and rearrange the equation to isolate desired variables such as illuminance \( E \).

By handling trigonometric identities, we unravel the relationships between various parts of the equation, providing insight into how angles affect light intensity and distribution. Grasping these identities makes tackling related mathematical problems more manageable.

• Helps in transforming complex equations.
• Involves identities like \( \sec \theta = \frac{1}{\cos \theta} \).
• Aids in isolating variables for solutions.