Problem 74 - Algebra and Trigonometry

Question

Pupil Size When the brightness x of a light source is increased, the eye reacts by decreasing the radius R of the pupil. The dependence of R on x is given by the function $$ R(x)=\sqrt{\frac{13+7 x^{0.4}}{1+4 x^{0.4}}} $$ where \(R\) is measured in millimeters and \(x\) is measured in appropriate units of brightness. (a) Find \(R(1), R(10),\) and \(R(100)\) (b) Make a table of values of \(R(x)\)

Step-by-Step Solution

Answer

\( R(1) = 2 \), \( R(10) \approx 1.664 \), \( R(100) \approx 1.475 \). Table created for \( x \) and \( R(x) \).

Step 1: Calculate \( R(1) \)

To find \( R(1) \), substitute \( x = 1 \) into the function:\[ R(1) = \sqrt{\frac{13 + 7 \cdot 1^{0.4}}{1 + 4 \cdot 1^{0.4}}} = \sqrt{\frac{13 + 7 \cdot 1}{1 + 4 \cdot 1}} = \sqrt{\frac{13 + 7}{1 + 4}} = \sqrt{\frac{20}{5}} = \sqrt{4} = 2. \]

Step 2: Calculate \( R(10) \)

To find \( R(10) \), substitute \( x = 10 \) into the function:\[ R(10) = \sqrt{\frac{13 + 7 \cdot 10^{0.4}}{1 + 4 \cdot 10^{0.4}}} = \sqrt{\frac{13 + 7 \cdot 2.5118864315}{1 + 4 \cdot 2.5118864315}}. \]Approximately, \( 10^{0.4} \approx 2.5118864315 \).Substitute values: \[ R(10) = \sqrt{\frac{13 + 7 \times 2.5118864315}{1 + 4 \times 2.5118864315}} \approx \sqrt{\frac{13 + 17.58320402}{1 + 10.04754573}} \approx \sqrt{\frac{30.58320402}{11.04754573}} \approx \sqrt{2.768976419} \approx 1.664. \]

Step 3: Calculate \( R(100) \)

To find \( R(100) \), substitute \( x = 100 \) into the function:\[ R(100) = \sqrt{\frac{13 + 7 \cdot 100^{0.4}}{1 + 4 \cdot 100^{0.4}}} = \sqrt{\frac{13 + 7 \cdot 6.3095734448}{1 + 4 \cdot 6.3095734448}}. \]Approximately, \( 100^{0.4} \approx 6.3095734448 \).Substitute values: \[ R(100) = \sqrt{\frac{13 + 7 \times 6.3095734448}{1 + 4 \times 6.3095734448}} \approx \sqrt{\frac{13 + 44.16701361}{1 + 25.23829378}} \approx \sqrt{\frac{57.16701361}{26.23829378}} \approx \sqrt{2.179} \approx 1.475. \]

Step 4: Create a Table of Values

Now that we have computed values for specific \( x \), we can create a table to list \( x \) and \( R(x) \) values:| x | \( R(x) \) ||-----|--------|| 1 | 2 || 10 | 1.664 || 100 | 1.475 |This table summarizes the pupil's radius at different brightness levels.

Key Concepts

Pupil Size

The size of the pupil is the aperture through which light enters the eye. It is critical in regulating the amount of light that reaches the retina. The pupil can expand or contract to respond to different lighting conditions. When the surrounding environment is dark, the pupil dilates, allowing more light to enter. Conversely, it constricts under bright conditions to limit light exposure.

• **Role:** Controls the light entering the eye, ensuring optimal vision.
• **Mechanism:** Changes size in response to light intensity.

The mathematical model in this exercise describes this natural adjustment in pupil size using a function. Understanding the pupil's behavior in various light conditions helps us appreciate how the eye adapts to protect itself from damage while maximizing its ability to perceive images.

Dependency on Light Brightness

The radius of the pupil depends heavily on the brightness of the light source. In the given function, brightness is represented by the variable \(x\). As the value of \(x\) increases, it means the light is getting brighter.
To ensure that the eye is protected from excessive brightness, the pupil size reduces, shown by our calculated decreases in \(R(x)\) as \(x\) grows:

• **R(1):** When brightness is minimal (close to 1), the pupil's radius is larger (2 mm).
• **R(10):** For moderate brightness, the pupil diminishes (1.664 mm).
• **R(100):** Under very bright conditions, the pupil further constricts (1.475 mm).

This relationship is crucial because it highlights the ability of the human eye to adapt and protect itself from harm by adjusting the light intake based on different brightness levels.

Mathematical Function Evaluation

Evaluating a function is like following a recipe to find the specific output for a particular input. Here, we have a function for the pupil's radius: \[ R(x) = \sqrt{\frac{13 + 7x^{0.4}}{1 + 4x^{0.4}}} \] This function denotes how the radius of the pupil changes with light brightness. To evaluate the pupil size at particular brightness levels like \(x = 1\), \(x = 10\), and \(x = 100\), we plug these values into the function:

• **Substitute:** Place the brightness value into the function.
• **Calculate:** Perform the operations: exponentiation, multiplication, and division.
• **Simplify:** Find the square root of the result to get the pupil size.

For instance, substituting \(x = 1\) gives us a straightforward computation resulting in a simpler form and a larger radius of 2 mm, while \(x = 100\) requires handling larger numbers, ultimately showing more contraction in the pupil size (1.475 mm). These steps help us understand how theoretical math applies to real physiological phenomena.