Problem 56 - College Algebra with Corequisite Support

Question

For the following exercises, use the given information to answer the questions. The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 foot-candles at a distance of 3 meters. Find the intensity level at 8 meters.

Step-by-Step Solution

Answer

The intensity at 8 meters is 0.01125 foot-candles.

Step 1: Understand the Relationship

The problem states that light intensity varies inversely with the square of the distance. This implies the relationship can be stated as \( I = \frac{k}{d^2} \), where \( I \) is the intensity, \( d \) is the distance, and \( k \) is a constant to be determined.

Step 2: Determine the Constant

Use the given information \( I = 0.08 \) foot-candles at \( d = 3 \) meters. Substitute into the equation to find \( k \): \( 0.08 = \frac{k}{3^2} \), which simplifies to \( k = 0.08 \times 9 = 0.72 \).

Step 3: Apply the Constant to New Distance

Use the constant \( k = 0.72 \) and substitute \( d = 8 \) meters into the inverse variation equation \( I = \frac{k}{d^2} \). This gives \( I = \frac{0.72}{8^2} = \frac{0.72}{64} = 0.01125 \) foot-candles.

Key Concepts

Light Intensity

Light intensity is a measure of how much luminous energy hits a certain area. Think of it like how bright a lamp seems to your eyes. The closer you are to the lamp, the brighter the light feels. This brightness that you perceive is the light's intensity. In practical terms, light intensity is determined by the amount of illumination a surface receives.

When we talk about measuring light, we often use units like "candles" from back in the day when light was measured by the candlelight. Today, we use a more modern unit called "foot-candles" designed around practical measurement needs.

Distance and Intensity Relationship

The relationship between distance and intensity of light is quite fascinating. As you move away from a light source, the intensity doesn't just decrease gradually; it decreases a lot faster. Specifically, light intensity decreases with the square of the distance from the source.

This means if you double the distance from the light source, the intensity becomes one-fourth of what it was. This relationship is a clear example of inverse variation. In mathematical terms, if you have an equation like \( I = \frac{k}{d^2} \), then

• \( I \) is the intensity
• \( d \) is the distance
• \( k \) is a constant based on initial conditions

Understanding this concept is essential when dealing with tasks involving illumination and design.

Foot-candles

Foot-candles are a unit of measurement that describes how bright a light source appears when you stand one foot away from it. The term comes from the early times when light was measured with candles and their illuminating strength.
Today, architects and lighting designers commonly use foot-candles to determine needed lighting levels in various environments.

• They help decide how much light is necessary for different activities like reading or cooking.
• They are essential in creating optimal lighting conditions while conserving energy.

The use of foot-candles is especially crucial in situations where precise lighting is required, such as in museums or workspaces.

Mathematics Problem Solving

Mathematics problem solving involves using logical and analytical skills to solve challenges such as inverse variation problems. These skills become a way to unravel relationships like those between distance and intensity of light.

The first step is typically to understand the problem. Identify known elements, like intensity or distance, and relate them using formulas.

• Here, the relationship is modeled by \( I = \frac{k}{d^2} \).
• After establishing the relationship, use given data to find constants like \( k \), which personalize the equation to specific scenarios.
• Finally, apply your formula to new situations, like finding intensity at different distances.

This series of steps emphasizes the thought process and techniques for tackling similar real-world problems effectively using mathematics.