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Chapter 5: Problem 11

Represents an exponential function. Construct that function and then identify the corresponding growth or decay rate in percentage form. $$ \begin{array}{ccccc} \hline x & 0 & 1 & 2 & 3 \\ y & 225.00 & 228.38 & 231.80 & 235.28 \\ \hline \end{array} $$

Short Answer

Expert verified
The exponential function is \( y = 225.00 \times (1.015)^x \) with a growth rate of approximately 1.5%.

Step by step solution

01

Identify the general form

An exponential function can be generally represented as \( y = a \times b^x \)
02

Determine the initial value

When \(x = 0\), \(y = 225.00\). Therefore, the initial value, \(a\), is 225.00.
03

Calculate the base

To find the base, \(b\), use two points from the dataset. For example, when \(x = 0\), \(y = 225.00\) and when \(x = 1\), \(y = 228.38\). Then,\( 228.38 = 225.00 \times b \implies b = \frac{228.38}{225.00} \approx 1.015\)
04

Construct the exponential function

Now, substitute \(a\) and \(b\) into the general form,\( y = 225.00 \times (1.015)^x \)
05

Identify the growth rate

The growth rate can be found from the base, \(b\). The growth rate is the difference between \(b\) and 1, expressed as a percentage:\((1.015 - 1) \times 100\% \approx 1.5\%\)

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Key Concepts

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

Growth Rate
The growth rate in an exponential function represents how the function's value changes over time.
More specifically, it shows the percentage by which the value increases or decreases at each step.
In our problem, we determined the growth rate using the base of the exponential function, which we found to be about 1.015.
To convert this base into a growth rate, subtract 1 from the base and then multiply by 100 to get the percentage.
This calculation gave us a growth rate of 1.5%.
This means that for each unit increase in x, the value of y increases by 1.5%.
Understanding the growth rate is crucial for understanding the behavior of the function over time.
Initial Value
The initial value in an exponential function is the value of the function when the input (x) is zero.
In our example, we found that when x = 0, y = 225.00.
This tells us that the initial value, or starting point, of the function is 225.00.
The initial value is significant because it serves as the anchor point for the entire function.
It represents the function's value before any growth or decay has occurred.
In mathematical terms, the initial value is denoted as 'a' in the general form of an exponential function, which is y = a x b^x.
By knowing the initial value, you can better understand and predict the function's behavior at other points.
Base Calculation
The base of an exponential function is a crucial component that determines the rate and direction (growth or decay) of the function.
In our case, we need to find the base, denoted as b, to construct the function.
We calculated the base by using two given points from the dataset: (x=0, y=225.00) and (x=1, y=228.38).
By setting up the equation 228.38 = 225.00 x b and solving for b, we found the base to be approximately 1.015.
The base is significant because it shows how much the function's value changes for each unit increase in x.
In the case of a base greater than 1, the function experiences growth; if the base is less than 1, the function experiences decay.
Percentage Growth
Percentage growth helps quantify the growth rate in a way that is easy to understand and compare.
In exponential functions, the percentage growth is derived from the base.
For our problem, we calculated the base to be approximately 1.015, indicating an increase.
To convert the base to a percentage, subtract 1 (the neutral value) and multiply by 100.
This gives us the percentage growth, which in our case is about 1.5%.
This means the function's value grows by 1.5% for each unit increase in x.
Knowing the percentage growth allows you to understand how quickly the value of the function is increasing and make more informed predictions about future values.

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Most popular questions from this chapter

The populations of four towns for time \(t,\) in years, are given by: $$ \begin{array}{l} P_{1}(t)=12,000(1.05)^{t} \\ P_{2}(t)=6000(1.07)^{t} \\ P_{3}(t)=100,000(1.01)^{t} \\ P_{4}(t)=1000(1.9)^{t} \end{array} $$ a. Which town has the largest initial population? b. Which town has the largest growth factor? c. At the end of 10 years, which town would have the largest population?

Generate quick sketches of each of the following functions, without the aid of technology. $$ \begin{array}{ll} f(x)=4(3.5)^{x} & g(x)=4(0.6)^{x} \\ h(x)=4+3 x & k(x)=4-6 x \end{array} $$ a. As \(x \rightarrow+\infty\), which function(s) approach \(+\infty ?\) b. As \(x \rightarrow+\infty\), which function(s) approach \(0 ?\) c. As \(x \rightarrow-\infty\), which function(s) approach \(-\infty\) ? d. As \(x \rightarrow-\infty\), which function(s) approach 0 ?

Write the equation of each exponential growth function in the form \(y=C a^{x}\) where: a. The initial population is 350 and the growth factor is \(\frac{4}{3}\). b. The initial population is \(5 \cdot 10^{9}\) and the growth factor is \(1.25 .\) c. The initial population is 150 and the population triples during each time period. d. The initial population of 2 quadruples every time period.

Construct both a linear and an exponential function that go through the points (0,200) and (10,500) .

(Requires technology to find a best-fit function.) Estimates for world population vary, but the data in the accompanying table are reasonable estimates. $$ \begin{aligned} &\text { World Population }\\\ &\begin{array}{cc} \hline & \text { Total Population } \\ \text { Year } & \text { (millions) } \\ \hline 1800 & 980 \\ 1850 & 1260 \\ 1900 & 1650 \\ 1950 & 2520 \\ 1970 & 3700 \\ 1980 & 4440 \\ 1990 & 5270 \\ 2000 & 6080 \\ 2005 & 6480 \\ \hline \end{array} \end{aligned} $$ a. Enter the data table into a graphing program (you may wish to enter 1800 as 0,1850 as \(50,\) etc. ) or use the data file WORLDPOP in Excel or in graph link form. b. Generate a best-fit exponential function. c. Interpret each term in the function, and specify the domain and range of the function. d. What does your model give for the growth rate? e. Using the graph of your function, estimate the following: i. The world population in \(1750,1920,2025,\) and 2050 ii. The approximate number of years in which world population attained or will attain 1 billion (i.e., 1000 million), 4 billion, and 8 billion f. Estimate the length of time your model predicts it takes for the population to double from 4 billion to 8 billion people.
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