/*! 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 13 Suppose a colony of bacteria has... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 13

Suppose a colony of bacteria has a continuous growth rate of \(30 \%\) per hour. If the colony contains 8000 cells now, how many did it contain five hours ago?

Short Answer

Expert verified
The bacterial colony contained approximately 29,703 cells five hours ago.

Step by step solution

01

Identify given information

We are given the following information: - Growth rate (\(r\)): 30%, which can be converted to a decimal as 0.30 - Current population (\(P_t\)): 8000 cells - Time period (\(t\)): 5 hours ago (we will use a negative value for time since we are going back in time)
02

Convert the growth rate to a decimal

We need to convert the given growth rate from a percentage to a decimal for calculations. To do this, divide the percentage by 100: \[r = \frac{30}{100} = 0.30\]
03

Insert values into exponential growth formula and solve

Using the exponential growth formula, we have \(P_t = P_0 \cdot (1 + r)^t\). Our goal is to find the initial population (\(P_0\)), so we will rearrange the equation to solve for \(P_0\). \[ \begin{aligned} P_t &= P_0 \cdot (1 + r)^t\\ \Rightarrow P_0 &= \frac{P_t}{(1+r)^t} \end{aligned} \] Now, substitute the given values into the equation: \[ \begin{aligned} P_0 &= \frac{8000}{(1 + 0.30)^{-5}}\\ P_0 &= \frac{8000}{(1.30)^{-5}}\\ P_0 &= 8000 \cdot (1.30)^{5}\\ P_0 &= 8000 \times 3.71293 \end{aligned} \]
04

Calculate the initial population

After substituting the given values, now calculate the initial population: \[ \begin{aligned} P_0 &= 8000 \times 3.71293\\ P_0 &= 29,703.44 \end{aligned} \] Since the number of bacterial cells must be a whole number, we can round the result to the nearest whole number: \(P_0 \approx 29,703\)
05

Conclusion

The bacterial colony contained approximately 29,703 cells five hours ago.

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

Show that \(\sinh\) is a one-to-one function and that its inverse is given by the formula $$ (\sinh )^{-1}(y)=\ln \left(y+\sqrt{y^{2}+1}\right) $$ for every real number \(y\).

Puppose \(r\) is a small positive number. Estimate the slope of the line containing the points \(\left(7, e^{7}\right)\) and \(\left(7+r, e^{7+r}\right)\)

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

Show that sinh is an odd function.

Find a number \(c\) such that \(\ln c=5\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.