/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Suppose a colony of bacteria has... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 15

Suppose a colony of bacteria has a continuous growth rate of \(35 \%\) per hour. How long does it take the colony to triple in size?

Short Answer

Expert verified
The bacteria colony will take approximately \(t = \frac{\ln{3}}{0.35} \approx 3.16\) hours to triple in size.

Step by step solution

01

Determine the Exponential Growth Formula

The general formula for exponential growth is given by: \[P(t) = P_0e^{rt}\] Where: - \(P(t)\) is the population at time t - \(P_0\) is the initial population - \(r\) is the growth rate (expressed as a decimal) - \(t\) is the time (in hours, in this case)
02

Set up the equation with given information

Since the bacteria colony triples in size, we can represent the final population as 3\(P_0\). The continuous growth rate is 35% per hour, so the growth rate as a decimal is 0.35. Using this information, we will set the exponential growth formula like this: \[3P_0 = P_0e^{0.35t}\]
03

Solve for the variable t

To find the time it takes for the bacteria colony to triple, we need to solve for t. First, we need to cancel out the \(P_0\) on both sides of the equation: \[3 = e^{0.35t}\] Now take the natural logarithm (ln) of both sides of the equation: \[\ln{3} = \ln{e^{0.35t}}\] Apply the power rule for logarithms on the right side, which states that \(\ln{a^b}=b\ln{a}\): \[\ln{3} = 0.35t\ln{e}\] Since \(\ln{e}=1\): \[\ln{3} = 0.35t\] Finally, isolate the variable \(t\): \[t = \frac{\ln{3}}{0.35}\]
04

Calculate the time

Plug in the values and calculate the value of t: \[t = \frac{\ln{3}}{0.35} \approx 3.16\] The bacteria colony will take approximately 3.16 hours to triple in size.

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!

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

For \(x=3\) and \(y=8\), evaluate each of the following: (a) \(\ln (x y)\) (b) \((\ln x)(\ln y)\)

Find all numbers \(x\) that satisfy the given equation. \(\ln (x+5)+\ln (x-1)=2\)

For \(x=1.1\) and \(y=5\), evaluate each of the following: (a) \(\ln (x y)\) (b) \((\ln x)(\ln y)\)

Show that the range of cosh is the interval \([1, \infty)\).

Find a number \(w\) such that \(\ln (3 w-2)=5\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.