$Math Studyset Vaia Explanations$ Math

5 Edition

Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan

Chapter 3: Chapter 3

Problem 1

Next question

Given that a quantity $Q(t)$ is described by the exponential growth function $$ Q(t)=400 e^{\mathrm{a} .05 t} $$ where $t$ is measured in minutes, answer the following questions: a. What is the growth constant? b. What quantity is present initially? c. Complete the following table of values:

Short Answer

Expert verified

a. The growth constant is $0.05\mathrm{a}$. b. The initial quantity present is 400 units. c. The completed table of values is: | Time (t) | Quantity (Q) | |:--------------:|:----------------------------:| | 0 | 400 | | 5 | $400e^{0.25\mathrm{a}}$ | | 10 | $400e^{0.5\mathrm{a}}$ | | 15 | $400e^{0.75\mathrm{a}}$ |

See the step by step solution

Step by step solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Identify the growth constant

The growth constant is the coefficient of $t$ in the given exponent of the function: $\mathrm{a} \cdot 0.05$. Therefore, the growth constant is $0.05\mathrm{a}$.

Step 2: Determine the initial value

In order to determine the initial quantity present in the system, we must evaluate $Q(t)$ when $t=0$. That is: $$ Q(0) = 400e^{\mathrm{a} \cdot 0.05\cdot 0} = 400e^0 = 400 $$ So, initially, there are 400 units of the quantity present.

Step 3: Complete the table of values

We will now find the value of the function at different times $t$. We can organize these values in a table as shown below: | Time (t) | Quantity (Q) | |:--------------:|:----------------:| | 0 | 400 | | 5 | $400e^{0.05\cdot 5\cdot \mathrm{a}}$ | | 10 | $400e^{0.05\cdot 10\cdot \mathrm{a}}$ | | 15 | $400e^{0.05\cdot 15\cdot \mathrm{a}}$ | Using the given formula, we can calculate the value of $Q(t)$ for the given time values: For $t=5$ minutes: $$ Q(5) = 400e^{0.05\cdot 5\cdot \mathrm{a}} = 400e^{0.25\mathrm{a}} $$ For $t=10$ minutes: $$ Q(10) = 400e^{0.05\cdot 10\cdot \mathrm{a}} = 400e^{0.5\mathrm{a}} $$ For $t=15$ minutes: $$ Q(15) = 400e^{0.05\cdot 15\cdot \mathrm{a}} = 400e^{0.75\mathrm{a}} $$ The completed table of values is as follows: | Time (t) | Quantity (Q) | |:--------------:|:----------------------------:| | 0 | 400 | | 5 | $400e^{0.25\mathrm{a}}$ | | 10 | $400e^{0.5\mathrm{a}}$ | | 15 | $400e^{0.75\mathrm{a}}$ |

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Recommended explanations on Math Textbooks

Find Study Materials for

Create Study Materials

Select your language

Chapter 3: Chapter 3

Given that a quantity \(Q(t)\) is described by the exponential growth function $$ Q(t)=400 e^{\mathrm{a} .05 t} $$ where \(t\) is measured in minutes, answer the following questions: a. What is the growth constant? b. What quantity is present initially? c. Complete the following table of values:

Short Answer

Step by step solution

Unlock all solutions

Step 1: Identify the growth constant

Step 2: Determine the initial value

Step 3: Complete the table of values

Access millions of textbook solutions in one place

Most popular questions from this chapter

More chapters from the book ‘Applied Mathematics: For the Managerial, Life, and Social Sciences’

Chapter 1

Chapter 2

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Join over 22 million students in learning with our Vaia App

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Calculus

Statistics

Pure Maths

Geometry