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Chapter 0: Problem 13

As broadband Internet grows more popular, video services such as YouTube will continue to expand. The number of online video viewers (in millions) is projected to grow according to the rule $$N(t)=52 t^{0.531} \quad 1 \leq t \leq 10$$ where \(t=1\) corresponds to 2003 . a. Sketch the graph of \(N\). b. How many online video viewers will there be in \(2010 ?\)

Short Answer

Expert verified
a) The graph of \(N(t) = 52t^{0.531}\) is a smooth curve, which increases as t increases, with points (1, 52), (3, 80.36), (5, 101.96), (7, 120.24), and (10, 144.75). b) In 2010, there will be approximately 128.7 million online video viewers.

Step by step solution

01

a) Sketch the graph of \(N(t)\)

To sketch the graph of \(N(t) = 52t^{0.531}\), we need to plot some points on the graph and connect them smoothly to get a rough idea of the shape of the function. We will choose some values for \(t\) and then calculate their corresponding \(N(t)\) values: t | N(t) ----- 1 | 52(1)^{0.531} = 52 3 | 52(3)^{0.531} 鈮� 80.36 5 | 52(5)^{0.531} 鈮� 101.96 7 | 52(7)^{0.531} 鈮� 120.24 10 | 52(10)^{0.531} 鈮� 144.75 Now, plot these points on a graph and connect them smoothly to get the following graph: (Include a plot of the function with the points plotted for better visualization)
02

b) Calculate the number of online video viewers in 2010

To find the number of online video viewers in 2010, we need to find the value of \(t\) that corresponds to 2010. Since \(t = 1\) corresponds to 2003, we can find \(t\) for 2010 by adding the years difference: t(2010) = 1 + (2010 - 2003) = 1 + 7 = 8. Now, plug in this \(t\) value into the function \(N(t)\) to get the number of online video viewers in 2010: \(N(8) = 52(8)^{0.531} \approx 128.7\) million viewers. So, there will be approximately 128.7 million online video viewers in 2010.

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Key Concepts

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

Mathematical Modeling
Mathematical modeling is a powerful tool used to describe and analyze real-world situations using mathematical concepts. It transforms real-world problems into mathematical forms so that they can be studied and solved systematically. In our exercise, the growth in the number of online video viewers is a real-world scenario that can be modeled mathematically.

The formula given as \(N(t) = 52t^{0.531}\) is the mathematical model that represents the number of online video viewers, in millions, as a function of time \(t\), where \(t=1\) represents the year 2003. This model allows us to perform various analyses, such as predicting the number of viewers for a given year or understanding the rate of growth over time. Mathematical models like this are created through observation of trends and fitting mathematical functions that closely match the data.
Exponential Functions
Exponential functions are a type of mathematical function widely used to model growth or decay processes, where the rate of change is proportional to the current value. These functions generally take the form \(y = ab^x\), where \(a\) is the initial amount, \(b\) is the growth factor, and \(x\) is the exponent representing time or another variable.

In the context of the growth of online video viewers, the function \(N(t) = 52t^{0.531}\) approximates an exponential behavior where the number of viewers grows at a rate that gets faster over time. Although the exponent, 0.531, is not an integer, the function still follows a pattern of rapid growth indicative of many natural processes, including population growth and technological adoption.
Graphing Functions
Graphing functions is a visual way of representing the relationship between two variables, usually with one variable along the x-axis and another along the y-axis. A graph illustrates the behavior of the function and can be crucial for understanding, communicating, and predicting patterns.

To graph the function \(N(t)=52t^{0.531}\), we follow a process of choosing input values for \(t\), calculating the corresponding \(N(t)\), and then plotting these points on a coordinate plane. Connecting the points gives us a visual representation of how the number of online video viewers grows over time. From the sketched graph, we can see the curvature characteristic of exponential growth, confirming the acceleration in the growth of online video viewer numbers as time progresses.

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