Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

The half-life of bismuth-210, $ ^{210} Bi $, is 5…

03:02

Question

Answered step-by-step

Problem 30 Hard Difficulty

A bacteria culture starts with 500 bacteria and doubles in size every half hour.

(a) How many bacteria are there after 3 hours?
(b) How many bacteria are there after $ t $ hours?
(c) How many bacteria are there after 40 minutes?
(d) Graph the population function and estimate the time for the population to reach 100,000.


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Heather Zimmers
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Heather Zimmers

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

More Answers

02:52

Clarissa Noh

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Calculus: Early Transcendentals

Chapter 1

Functions and Models

Section 4

Exponential Functions

Related Topics

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

Discussion

You must be signed in to discuss.
Top Calculus 3 Educators
Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Michael Jacobsen

Idaho State University

Calculus 3 Courses

Lectures

Video Thumbnail

04:31

Multivariate Functions - Intro

A multivariate function is a function whose value depends on several variables. In contrast, a univariate function is a function whose value depends on only one variable. A multivariate function is also called a multivariate expression, a multivariate polynomial, a multivariate series, or a multivariate function of several variables.

Video Thumbnail

12:15

Partial Derivatives - Overview

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

Join Course
Recommended Videos

02:52

A bacteria culture starts …

01:09

A bacterial culture starts…

01:07

A bacterial culture starts…

04:21

A bacterial culture starts…

10:08

A bacterial culture starts…

06:02

A bacteria culture starts …

04:08

The number of bacteria in …

01:09

Under ideal conditions a c…

05:59

Bacteria Culture The count…

04:18

Bacteria Culture A culture…

09:15

Bacteria Culture
after …

Watch More Solved Questions in Chapter 1

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38

Video Transcript

all right in this problem, we have a bacteria population that's doubling, doubling every half an hour so we could make a little chart that represents the time that has elapsed on the number of bacteria. So what time? Zero. We're starting with 500 bacteria, half a Knauer goes by and now it's doubled to 1000. Another half on hour goes by and now there's 2000 and so on. It keeps doubling. We want to know the number of bacteria when three hours have elapsed. So we've doubled until we got to three hours and there are 32,000 bacteria. So that takes care of part A. Now what we want to do is generalize so that we can say how many bacteria there would be after t hours. So what we want to do is look at the numbers in the table and find the pattern. So the first number was just 500. The next number was 500 times two or we could say times two to the first. The next number was 500 times to the second power. Then we have 500 times, two to the third, etcetera, continuing on that pattern. So where do we get this number, This exponents here? Well, that would be the number of compounding Zor the number of double ings we've had. And if three hours have gone by and it doubles twice every hour than there have been six double ing's so we can generalize this and we come up with a formula 500 times to raise to the to T, where T is the number of hours so that will tell us how many bacteria we have at time. T. So that helps us with part B because Part B asks us how many bacteria are there after t hours, and the answer would be 500 times two to the power to teach or to use an hours. Now, let's use that formula to answer part. See how many bacteria are there after 40 minutes, while we're going to have to convert 40 minutes into hours. So 40 minutes is 2/3 of an hour, and now we can substitute that into our expression above 40. So we get 500 times to raise to the 2/3 times two and 2/3 times two is 4/3. We put that in a calculator and round, and we get approximately 1260 bacteria. And lastly, we want to graft this and look for the time when the population reaches 100,000. So using a calculator, we can go to y equals and we can type are function into the y equals menu, and then we can also type y equals 100,000 into the Y equals menu, and that will give us a horizontal line at a height of 100,000 that stands for 100,000 bacteria. So a window I've chosen for this is going from negative 1 to 6 on the X axis and negative 211,000 on my Y axis. You can choose different numbers here. Just fiddle with your window values until you find something you like. Make sure you go at least to 100,000. So here's the graph. The blue graph is that exponential curve, the one that models the number of bacteria and the red line is that line at a height of 100,000 and now we're going to look for where they intersect so we can go into the calculate menu. We can choose Number five Intersect. We can put the cursor on the first curve press, enter cursor on the second curve, press enter, and then we can move the cursor closer to the intersection point and press enter and looking at the bottom of the screen, we see that it's 3.82 so that would be about 3.82 hours until the population reaches 100,000.

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
142
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
68
Hosted by: Alonso M
See More

Related Topics

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

Top Calculus 3 Educators
Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

Michael Jacobsen

Idaho State University

Calculus 3 Courses

Lectures

Video Thumbnail

04:31

Multivariate Functions - Intro

A multivariate function is a function whose value depends on several variables. In contrast, a univariate function is a function whose value depends on only one variable. A multivariate function is also called a multivariate expression, a multivariate polynomial, a multivariate series, or a multivariate function of several variables.

Video Thumbnail

12:15

Partial Derivatives - Overview

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

Join Course
Recommended Videos

02:52

A bacteria culture starts with 500 bacteria and doubles in size every half hour…

01:09

A bacterial culture starts with 500 bacteria and doubles in size every half hou…

01:07

A bacterial culture starts with 500 bacteria and doubles in size every half hou…

04:21

A bacterial culture starts with 500 bacteria and doubles in size every half ho…

10:08

A bacterial culture starts with 500 bacteria and doubles in size every half ho…

06:02

A bacteria culture starts with 300 bacteria and doubles in size every half hour…

04:08

The number of bacteria in a culture is modeled by the function $$n(t)=500 e^{0…

01:09

Under ideal conditions a certain bacteria population is known to double every t…

05:59

Bacteria Culture The count in a culture of bacteria was 400 after 2 hours and $…

04:18

Bacteria Culture A culture starts with 8600 bacteria. After one hour the count …

09:15

Bacteria Culture after 2 hours and $25,600$ after 6 hours. $$ \begin{array…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started