Problem 1 The following exponential functi... [FREE SOLUTION]

91影视

Explorations in College Algebra

Linda Almgren Kime, Judy Clark, Beverly K. Michael

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 5: Problem 1

The following exponential functions represent population growth. Identify the initial population and the growth factor. a. $Q=275 \cdot 3^{T}$ b. $P=15,000 \cdot 1.04^{t}$ c. $y=\left(6 \cdot 10^{8}\right) \cdot 5^{x}$ d. $A=25(1.18)^{t}$ e. $P(t)=8000(2.718)^{t}$ f. $f(x)=4 \cdot 10^{5}(2.5)^{x}$

Short Answer

Expert verified

a. Initial = 275, Growth Factor = 3; b. Initial = 15,000, Growth Factor = 1.04; c. Initial = 600,000,000, Growth Factor = 5; d. Initial = 25, Growth Factor = 1.18; e. Initial = 8,000, Growth Factor = 2.718; f. Initial = 400,000, Growth Factor = 2.5.

Step by step solution

- Identify the Initial Population

For exponential functions of the form $y = a \cdot b^{x}$, the initial population is the coefficient $a$. Locate this coefficient in each of the functions provided.

- Identify the Growth Factor

In an exponential function $y = a \cdot b^{x}$, the base $b$ of the exponential term represents the growth factor. Identify this base in each function.

- Example Analysis

Using the steps above, analyze each equation separately to find the initial population and growth factor.

Step 4a - Analyze Function a

For the function $Q=275 \cdot 3^{T}$, the initial population is $275$ and the growth factor is $3$.

Step 4b - Analyze Function b

For the function $P=15,000 \cdot 1.04^{t}$, the initial population is $15,000$ and the growth factor is $1.04$.

Step 4c - Analyze Function c

For the function $y=6 \cdot 10^{8} \cdot 5^{x}$, the initial population is $6 \cdot 10^{8} = 600,000,000$ and the growth factor is $5$.

Step 4d - Analyze Function d

For the function $A=25 \cdot 1.18^{t}$, the initial population is $25$ and the growth factor is $1.18$.

Step 4e - Analyze Function e

For the function $P(t)=8000 \cdot 2.718^{t}$, the initial population is $8000$ and the growth factor is $2.718$.

Step 4f - Analyze Function f

For the function $f(x)=4 \cdot 10^{5} \cdot 2.5^{x}$, the initial population is $4 \cdot 10^{5} = 400,000$ and the growth factor is $2.5$.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Initial Population

In exponential growth models, the initial population refers to the population size at the beginning of the observation period. It is commonly represented by the coefficient in an exponential function of the form $ y = a \times b^x $. Here's a brief breakdown of identifying the initial population in some example functions:

For $ Q = 275 \times 3^T $, the initial population is $ 275 $.
In the function $ P = 15,000 \times 1.04^t $, the initial population is $ 15,000 $.
For $ y = (6 \times 10^8) \times 5^x $, the initial population is $ 6 \times 10^8 $, which is $ 600,000,000 $.
In $ A = 25 \times 1.18^t $, the coefficient $ 25 $ is the initial population.
For $ P(t) = 8,000 \times 2.718^t $, the initial population is $ 8,000 $.
In the function $ f(x) = 4 \times 10^5 \times 2.5^x $, the initial population is $ 4 \times 10^5 $, which equals $ 400,000 $.

Growth Factor

The growth factor in an exponential function determines how rapidly the population increases. This is represented by the base $ b $ in the general form $ y = a \times b^x $. Let's look at some examples:

In $ Q = 275 \times 3^T $, the growth factor is $ 3 $.
For $ P = 15,000 \times 1.04^t $, the growth factor is $ 1.04 $.
In $ y = (6 \times 10^8) \times 5^x $, the growth factor is $ 5 $.
For $ A = 25 \times 1.18^t $, the growth factor is $ 1.18 $.
In $ P(t) = 8,000 \times 2.718^t $, the growth factor is $ 2.718 $.
For $ f(x) = 4 \times 10^5 \times 2.5^x $, the growth factor is $ 2.5 $.

Exponential Functions

Exponential functions are mathematical expressions used to describe processes where quantities grow at rates proportional to their current values. They have a general form of $ y = a \times b^x $, where:

$ y $ is the dependent variable (e.g., population size).
$ a $ is the initial value (initial population).
$ b $ is the growth factor.
$ x $ is the independent variable (often representing time).

Exponential functions are used in various fields such as biology, finance, and physics to model phenomena that exhibit rapid growth or decay.

Algebraic Functions

While exponential functions deal with rapidly growing or decaying quantities, algebraic functions involve polynomial expressions where variables are raised to whole number powers. For instance, a quadratic function like $ y = ax^2 + bx + c $ is an algebraic function. These functions often depict slower, more gradual changes compared to exponential growth. Understanding the key differences between algebraic and exponential functions is crucial for proper application in problem-solving and population modeling.

Population Modeling

Population modeling involves the use of mathematical functions to represent how populations change over time. Exponential growth models are especially common in scenarios where a population grows without significant constraints. The general form $ y = a \times b^x $ provides a straightforward means of predicting future population sizes based on current growth trends.

Initial population $ a $ sets the starting point for the model.
Growth factor $ b $ influences the rate of increase.

These models help in various fields, including ecology, economics, and social sciences, to make informed decisions based on predicted population trends.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.