Pupil Size When the brightness x of a light source is...

Pupil Size When the brightness $x$ of a light source is increased, the eye reacts by decreasing the Tadius $R$ 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 $R(x)$ c. Find the net change in the radius $R$ as $x$ changes from 10 to 100 .

Pupil Size When the brightness $x$ of a light source is increased, the eye reacts by decreasing the Tadius $R$ 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 $R(x)$
c. Find the net change in the radius $R$ as $x$ changes from 10 to 100 .

Key Concepts

Function Evaluation

This concept involves substituting specific values into a mathematical function to calculate its output. In the context of this problem, evaluating the function R(x) for given brightness levels (x = 1, 10, and 100) means replacing x with these values in the function's formula and simplifying the result. This process is fundamental for understanding how the output of a model depends on its input.

Creating a Table of Values

A table of values organizes computed function outputs corresponding to various inputs in a clear, structured form. This method provides a snapshot of the function's behavior over an interval, making it easier to observe trends, analyze patterns, and graph the relationship between brightness and pupil size.

Net Change in a Function

Net change refers to the difference between the function's values at two distinct points. In this problem, it involves calculating the difference in the pupil radius when the brightness changes from one level to another (from x = 10 to x = 100). This concept is essential in understanding how a quantity varies over an interval, a fundamental aspect of both algebra and calculus.

Transcript

00:01 We have been given a equation that rx here x is brightness, brightness of light source and rx is nothing but the radius of the pupil.

00:22 And we have been given how the radius of the pupil varies with brightness.

00:26 S.

00:29 This rx has been given as root over of 13 plus 7 x raised to the power 0 .4 divided by 1 plus 4x again raised to the power 0 .4.

00:46 Now this particular thing gives the values of rx for different values of x and in the question we have been asked to find what is r1.

00:54 So we will just putting 13.

00:58 Plus 7 and this become 1 because 1 raised to the power anything that is 0 .4 will be again your 1 divided by 1 plus 4 multiplied by 1 so what this gives this gives us value as root 13 plus 7 that is 20 and divided by 5 that is nothing but your 2 hence r1 comes out to be 2 similarly, proceeding for r10, we just put the values here 13 plus 7 multiplied by 10 raised to the power 0 .4 divided by 1 plus 4 multiplied by 10 raised to the power 0 .4.

01:42 Now this gives the value as 1 .664.

01:47 Similarly, if you move for r100, we get the equation something like, 13 plus 7 multiplied by 100 to the power 0 .4 divided by 1 plus 4 multiplied by 100 to the rest 0 .4.

02:09 And again the value comes up to be 1 .476.

02:14 Now these 3 values are for different values of r.

02:18 Now in the second part we have been asked to make a table of rx okay...

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