Problem 40 - 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. Is the equation in Exercise 39 equivalent to \(R^{2}=\frac{I \tan \theta \cos \theta}{E} ?\) Explain.

Step-by-Step Solution

Answer

No, unless \( I \tan \theta = 1 \).

Step 1: Understand the Given Equation

The given equation is \( \sec \theta = \frac{1}{ER^2} \). This relates the illuminance \( E \), distance \( R \), and angle \( \theta \).

Step 2: Transform the Secant Function

Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, the given equation can be expressed in terms of cosine as \( \frac{1}{\cos \theta} = \frac{1}{ER^2} \).

Step 3: Cross Multiply to Simplify

Cross multiply to get rid of the fractions: \( ER^2 = \cos \theta \).

Step 4: Solve for \( R^2 \)

Divide both sides by \( E \) to solve for \( R^2 \): \( R^2 = \frac{\cos \theta}{E} \).

Step 5: Compare with Given Equation

The equation we derived is \( R^2 = \frac{\cos \theta}{E} \), whereas the equation given is \( R^2 = \frac{I \tan \theta \cos \theta}{E} \). They are not the same unless \( I \tan \theta = 1 \).

Key Concepts

Light Intensity

Light intensity is a critical factor in determining the illuminance, which describes how much light falls on a particular area.
A point light source emits light in all directions, and the intensity is measured in candles. Intensity is a property of the light source itself and describes how much energy it emits over a unit of time.

• Consider a dull light bulb emitting lower intensity than a bright lamp.
• Intensity impacts the amount of light that can reach a surface, especially at greater distances.

Light intensity is a fundamental concept in lighting design, affecting everything from home interiors to public spaces to photography. Understanding light intensity helps ensure that environments have the correct lighting for comfort and functionality.
The relationship between light intensity and illuminance involves the distance from the light source, which is inversely proportional to the square of that distance and the cosine of the angle relative to the surface.
These mathematical connections ensure that light is calculated accurately, optimizing visibility and ambiance.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and secant, play a vital role in measuring angles and distances in physics and engineering.
The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function. Thus, \( \sec \theta = \frac{1}{\cos \theta} \).
This concept is often used in illuminance calculations where the angle \( \theta \) between the light beam and the surface is crucial.

• Changes in the angle directly affect the illuminance. A larger \( \theta \) increases the value of secant, reflecting more angled light delivery.
• Trigonometry allows converting between functions, helping simplify complex equations.

A profound understanding of these functions enables you to handle real-world problems related to light, such as maximizing efficiency by adjusting the angle of light fixtures.

Mathematical Equations

Mathematical equations are tools used to represent and solve physical phenomena like light distribution. In illuminating contexts, they equate various factors such as illuminance \( E \), light intensity \( I \), distance \( R \), and angle \( \theta \).
In the given context, the equation \( \sec \theta = \frac{1}{ER^2} \) encompasses multiple variables working together to explain how light behaves when it strikes a surface.

• Breaking down the equation requires algebraic manipulation, such as cross-multiplying to eliminate fractions.
• Transform the equation using properties of functions, like changing secant to cosine.
• Comparing different equation forms offers insight into whether adjustments in variables lead to consistent results.

In the problem scenario, understanding these mathematical equations allows students to explore how different conditions affect the light's impact, ensuring it can be measured and applied accurately.