Problem 78 - A Graphical Approach to College Algebra

Question

Solve each problem. The illumination \(I\) in foot-candles produced by a light source is related to the distance \(d\) in feet from the light source by the equation $$d=\sqrt{\frac{k}{I}}$$ where \(k\) is a constant. If \(k=400,\) how far from the source will the illumination be 14 foot-candles? Round to the nearest hundredth of a foot.

Step-by-Step Solution

Answer

The distance is approximately 5.34 feet.

Step 1: Substitute the Given Values

The given equation is \(d = \sqrt{\frac{k}{I}}\). We need to find \(d\) when \(k = 400\) and \(I = 14\). Substitute these values into the equation: \[d = \sqrt{\frac{400}{14}}\].

Step 2: Evaluate the Fraction

Calculate the fraction under the square root. Divide 400 by 14: \(\frac{400}{14} \approx 28.57\).

Step 3: Calculate the Square Root

Find the square root of the fraction: \(\sqrt{28.57} \approx 5.34\).

Step 4: Round the Result

Round the calculated distance to the nearest hundredth of a foot. So, \(d \approx 5.34\).

Key Concepts

Foot-Candles

Foot-candles are a unit of measurement that quantify the amount of light that hits a surface. Imagine standing in a room with a light bulb illuminating a table; the brightness you see on the table is measured in foot-candles. This unit is vital because it helps us understand how much light is needed for various tasks, like reading or working in a brightly lit area. To give you a sense of context:

• 1 foot-candle is roughly equivalent to the light of a candle shining over a 1-foot square area.
• It is a direct measure of lumens per square foot, where lumens tell us about the total amount of visible light.

Understanding foot-candles helps in designing spaces with adequate lighting to ensure comfort and usability. For instance, a living room might require lower foot-candles, while a workspace needs much more to avoid eye strain.

Distance from Light Source

The distance from the light source significantly affects how much light reaches a surface. You can imagine how a bulb lights up an area: closer proximity means brighter light, while increasing the distance dims the light. When calculating illumination levels in foot-candles, knowing the distance from the light source to the spot in question is crucial. This is because light disperses, or spreads out, as it travels. Here’s why distance matters:

• The further the light has to travel, the more it spreads out, reducing the light intensity.
• Accurately measuring the distance allows for precise lighting designs.

By controlling the distance, you can tweak how much light fills a room, maintaining the desired illumination level.

Inverse Square Law

The inverse square law is a fundamental principle to understand in illumination calculations. It explains how intensity changes with distance.The idea is simple: as light gets further from its source, it spreads out and decreases in intensity. The law states that illumination decreases with the square of the distance, which means doubling the distance reduces light intensity to one fourth.The mathematical relationship is outlined as:\[ I \propto \frac{1}{d^2} \]Where:

• \(I\) is the illumination or light intensity.
• \(d\) is the distance from the light source.

Understanding this is key when designing environments to ensure even light distribution. This principle is crucial for anyone working with lighting, from photographers to engineers designing road lighting.

Algebraic Equation Solving

Algebraic equation solving is a crucial skill in math where we find the unknown values that make the equation true. This often involves operations like substitution, simplification, or rearrangement.For our illumination problem, the equation given was:\[ d = \sqrt{\frac{k}{I}} \]Our task was to solve for \(d\) when the values of \(k\) and \(I\) are given. Here’s the step-by-step solution:

• Substitute the known values into the equation.
• Evaluate the fraction inside the square root for clarity.
• Calculate the square root to solve for \(d\).
• Round to the nearest hundredth to ensure practicality.

By following these steps, we dealt with mathematical operations methodically, just like solving puzzles.