Problem 64 - 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 $$ R(x)=\sqrt{\frac{13+7 x^{0.4}}{1+4 x^{0.4}}} $$ (a) Find \(R(1), R(10),\) and \(R(100)\) (b) Make a table of values of \(R(x)\)

Step-by-Step Solution

Answer

\( R(1) = 2 \); \( R(10) \approx 1.664 \); \( R(100) \approx 1.475 \); see table for values.

Step 1: Plug in x=1 into R(x)

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

Step 2: Plug in x=10 into R(x)

To find \( R(10) \), substitute \( x = 10 \) into the function:\[ R(10) = \sqrt{\frac{13 + 7 \cdot 10^{0.4}}{1 + 4 \cdot 10^{0.4}}} \]Calculate \( 10^{0.4} \) which is approximately 2.5118, then \[ R(10) = \sqrt{\frac{13 + 7 \cdot 2.5118}{1 + 4 \cdot 2.5118}} = \sqrt{\frac{13 + 17.5826}{1 + 10.0472}} = \sqrt{\frac{30.5826}{11.0472}} \]Calculate the division and then its square root, \[ R(10) \approx \sqrt{2.7689} \approx 1.664 \]

Step 3: Plug in x=100 into R(x)

To find \( R(100) \), substitute \( x = 100 \) into the function:\[ R(100) = \sqrt{\frac{13 + 7 \cdot 100^{0.4}}{1 + 4 \cdot 100^{0.4}}} \]Calculate \( 100^{0.4} \) which is approximately 6.3096, then \[ R(100) = \sqrt{\frac{13 + 7 \cdot 6.3096}{1 + 4 \cdot 6.3096}} = \sqrt{\frac{13 + 44.1672}{1 + 25.2384}} = \sqrt{\frac{57.1672}{26.2384}} \]Calculate the division and then its square root, \[ R(100) \approx \sqrt{2.1794} \approx 1.475 \]

Step 4: Create a table of values for R(x)

To make a table of values, use the function \( R(x) = \sqrt{\frac{13 + 7x^{0.4}}{1 + 4x^{0.4}}} \) and calculate it for desired values of \( x \), not just \( x = 1, 10, 100 \). Here are calculations for several values to demonstrate:- \( x = 1 \), \( R(1) \approx 2 \)- \( x = 5 \), calculate \( 5^{0.4} \approx 1.9037 \), proceed similarly to get \( R(5) \approx 1.817 \)- \( x = 10 \), \( R(10) \approx 1.664 \)- \( x = 50 \), calculate \( 50^{0.4} \approx 4.216 \), proceed similarly to get \( R(50) \approx 1.537 \)- \( x = 100 \), \( R(100) \approx 1.475 \)From the function, create a table:| \( x \) | \( R(x) \) ||------|-----|| 1 | 2 || 5 | 1.817 || 10 | 1.664 || 50 | 1.537 || 100 | 1.475 |

Key Concepts

Function Evaluation

Function evaluation is a simple yet crucial concept in mathematics. It involves substituting a specific value into a function to compute the result. This is essential for understanding how a function behaves for different inputs. In our exercise, we have a function that describes how the radius of a pupil changes with light brightness: \[ R(x) = \sqrt{\frac{13 + 7x^{0.4}}{1 + 4x^{0.4}}} \]To evaluate this function:- You substitute a value for \( x \) into the function, replacing any occurrence of \( x \) with that number.- Then, perform the arithmetic operations step by step, always following the order of operations (parentheses, exponents, multiplication and division, addition and subtraction). For example, to find \( R(1) \), replace \( x \) with 1 in the function. Calculate within the parentheses first, then move to the division and finally the square root. This process helps you understand a function's output based on different input values.

Mathematical Modeling

Mathematical modeling is the art of translating real-world scenarios into mathematical language. It provides a way to analyze and solve complex real-life problems by using mathematical structures and processes.In this exercise, the relationship between the brightness of light and pupil size is modeled by the function \( R(x) = \sqrt{\frac{13 + 7x^{0.4}}{1 + 4x^{0.4}}} \). This model:

• Uses the variable \( x \) to represent light brightness.
• Utilizes mathematical operations such as exponentiation and division to mimic the biological response of the eye to light.
• Results in a function \( R(x) \) that outputs the pupil's radius based on the input brightness \( x \).

This kind of model is powerful. It allows predictions about how the pupil will adapt under various lighting conditions, which is crucial for applications in fields such as ophthalmology and vision sciences.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the backbone of algebra and many mathematical computations.The function given in the exercise, \( R(x) = \sqrt{\frac{13 + 7x^{0.4}}{1 + 4x^{0.4}}} \), is an algebraic expression itself. It includes:

• Constants (e.g., 13 and 7) which are fixed values.
• Variables (\( x \)) which are symbols representing quantities that can change.
• Operations like exponentiation (\( x^{0.4} \)) and square roots, which transform the values of \( x \).

Understanding the structure of algebraic expressions is crucial because it dictates how calculations are performed. Knowing the role of each component helps in simplifying, rearranging, or solving these expressions. It also frames the systematic approach needed to analyze functions like \( R(x) \), finding them for multiple values, or even graphing their corresponding curves.