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Chapter 1: Problem 23

Many physical quantities are connected by inverse square laws, that is, by power functions of the form \(f(x)=k x^{-2}\) . In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light?

Short Answer

Expert verified
The light is 4 times brighter when you move halfway to the lamp.

Step by step solution

01

Understand the Inverse Square Law

The problem describes a situation where light intensity follows an inverse square law. Specifically, if the distance from the light source is doubled, the light intensity is reduced by a factor of four. Conversely, if the distance is halved, the light intensity increases.
02

Set Up the Initial Function

Let's assume that initially, you are at a distance of 2 units (for simplicity). The illumination function is given as \( f(x) = k x^{-2} \), where \( k \) is a proportionality constant and \( x \) is the distance. Thus, the initial intensity is \( f(2) = k (2)^{-2} = k/4 \).
03

Calculate New Illumination when Distance is Halved

When the distance is halved, you are now 1 unit away from the lamp. Plugging this new distance into the function, you get \( f(1) = k (1)^{-2} = k \).
04

Compare the Change in Illumination

Initially, the illumination is \( k/4 \) and after moving closer, it becomes \( k \). To find how much brighter the light is, compare the final intensity to the initial intensity. The light is now \( k / (k/4) = 4 \) times brighter.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Functions
Power functions are expressions where a variable is raised to a constant power. They take a form like this: \( f(x) = kx^{n} \). Here, \( k \) is a constant, and \( n \) is the power to which the variable \( x \) is raised. In relation to the inverse square law, the power \( n \) is -2, so the function becomes \( f(x) = kx^{-2} \). The important characteristic of power functions is how they can model relationships where one quantity changes at a rate proportional to the power of another quantity.
Light Intensity
Light intensity refers to the amount of light that is received on a surface. It measures how bright or dim a light source appears from a certain point. This is crucial when determining how well-illuminated an area is for tasks like reading. To calculate light intensity, we often use the inverse square law. It relates intensity to distance by stating that intensity changes with the square of the distance from the source. Hence, if you move closer to or further away from a light source, the intensity encountered changes significantly.
Distance and Illumination
The relationship between distance and illumination can seem a bit counterintuitive at first, but it's actually quite straightforward with the concept of the inverse square law. As you move closer to a light source, the distance decreases, leading to greater illumination. Conversely, as the distance increases, the light becomes dimmer. For instance, in our scenario, when you move halfway towards a lamp, the distance reduces by half, which results in light intensity increasing by four times. This is because the illumination is calculated as \( f(x) = kx^{-2} \), and when \( x \) is halved, \( x^{-2} \) becomes four times its previous value.
Proportionality Constant
The proportionality constant, denoted by \( k \) in inverse square laws, helps describe the specific relationship between distance and intensity for different lights. It varies based on the intrinsic properties of the light source, like its actual brightness or energy output. In the function \( f(x) = kx^{-2} \), \( k \) ensures that the calculated intensity reflects how the light source illuminates an object at a particular distance. While \( k \) may be unknown, its role is crucial to accurately compute changes in illumination as a result of distance alterations, like moving closer to a lamp.

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Most popular questions from this chapter

Drinking and driving In a medical study, researchers measured the mean blood alcohol concentration (BAC) of eight fasting adult male subjects (in mg/mL) after rapid consumption of 30 \(\mathrm{mL}\) of ethanol (corresponding to two standard alcoholic drinks). The BAC peaked after half an hour and the table shows measurements starting after an hour. $$\begin{array}{|c|c|c|c|c|c|}\hline t \text { hours) } & {1.0} & {1.25} & {1.5} & {1.75} & {2.0} \\ \hline \mathrm{BAC} & {0.33} & {0.29} & {0.24} & {0.22} & {0.18} \\ \hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|c|}\hline \text { t(hours) } & {2.25} & {2.5} & {3.0} & {3.5} & {4.0} \\ \hline \mathrm{BAC} & {0.15} & {0.12} & {0.069} & {0.034} & {0.010} \\ \hline\end{array}$$ (a) Make a scatter plot and a semilog plot of the data. (b) Find an exponential model and graph your model with the scatter plot. Is it a good fit? (c) Use your model and logarithms to determine when the BAC will be less than 0.08 \(\mathrm{mg} / \mathrm{mL}\) , the legal limit for driving.

Amplifying DNA Polymerase Chain Reaction (PCR) is a biochemical technique that allows scientists to take tiny samples of DNA and amplify them into large samples that can then be examined to determine the DNA sequence. (This is useful, for example, in forensic science.) The process works by mixing the sample with appropriate enzymes and then heating it until the DNA double helix separates into two individual strands. The enzymes then copy each strand, and once the sample is cooled the number of DNA molecules will have doubled. By repeatedly performing this heating and cooling process, the number of DNA molecules continues to double every temperature cycle (referred to as a PCR cycle). (a) Suppose a sample containg \(x\) molecules is collected from a crime scene and is amplified by PCR. Express the number of DNA molecules as a function of the number \(n\) of \(\mathrm{PCR}\) cycles. (b) There is a detection threshold of \(T\) molecules below which no DNA can be seen. Derive an equation for the number of PCR cycles it will take for the DNA sample to reach the detection threshold. (c) One way scientists determine the abundance of different DNA molecules in a sample is by measuring the difference in time it takes to reach the detection threshold for each. Sketch a graph of the number of cycles needed to reach the detection threshold as a function of the initial number of molecules. Comment on the relationship between differences in initial number of molecules and differences in the time to reach the detection threshold.

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(a) Find the domain of \(f(x)=\ln \left(e^{x}-3\right)\) (b) Find \(f^{-1}\) and its domain.
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